Thermal Quantum Field Theory - uni-frankfurt.deendrodi/thermalFT.pdf · 2020-06-02 · Quantum mechanics Let us consider the quantum mechanics of a particle of mass , moving in a

Thermal Quantum Field Theory

Gergely Endrődi

Version: February 8, 2018

Contents

Preface: Ising model 4

1 Quantum mechanics 7

1.1 Basis states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2 Path integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 The harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.1 Gaussian integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5 Summary for the harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Real scalar fields 13

2.1 Path integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Proper time representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5 High-temperature expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17


3 Complex scalar fields 21

3.1 Grand canonical ensemble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Path integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3 Bose-Einstein condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3.1 Thermal density at high temperature . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3.2 Finiteness of the thermal density at the onset . . . . . . . . . . . . . . . . . . . 25

3.3.3 Critical temperature and order of the transition . . . . . . . . . . . . . . . . . . 26

3.3.4 Spontaneous symmetry breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4 Proper time representation and renormalization . . . . . . . . . . . . . . . . . . . . . . 28


3.6 Charged scalars in lower dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 Fermion fields 32

4.1 Fermionic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.1.1 Energy representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.1.2 Path integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.2 Path integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3 Proper time representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.4 Fermionic and bosonic thermal sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2

CONTENTS 3

4.5 Equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.6 Nonzero chemical potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5 Background electromagnetic fields 40

5.1 Path integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.2 Minkowskian and Euclidean space-times . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.3 Fermions in a magnetic eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.4 Proper time representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435.5 Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.6 Magnetic susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.7 Schwinger eect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.8 Bosons in a magnetic eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.9 Meissner eect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

A Special functions and complex analysis 52

A.1 Γ-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52A.2 𝜁-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53A.3 Elliptic Θ-function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54A.4 Mellin transform and contour integration . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Bibliography 57

Preface

Ising model

As an appetizer for the theory of quantum elds at nonzero temperature, we begin with the simplestèld theory', the classical Z(2) eld theory. This is the Ising model, which we dene here on atwo-dimensional grid with sites 𝑖 = (𝑖𝑥, 𝑖𝑦). The number 𝑁 of sites is the volume of the system. Thevariables are the spins 𝑠𝑖 = ±1 at each point, and their dynamics is described by the Hamiltonian

𝐻(𝑠⌋ = −𝐽 ∑∐𝑖𝑗

𝑠𝑖𝑠𝑗 − ℎ∑𝑖

𝑠𝑖 . (1)

The rst term is a nearest-neighbor interaction that, for 𝐽 > 0, aims to align neighboring spins. Thesecond term mimics an external magnetic eld ℎ that aims to set all spins positive. A given set 𝑠of spins will be called a configuration. The probability of a conguration 𝑠 at nite temperature𝑇 = 1⇑𝛽 is

𝑃 (𝑠) = 𝑒−𝛽𝐻(𝑠⌋ , (2)

and the partition function is the sum over all congurations:

𝒵 =∑𝑠

𝑒−𝛽𝐻(𝑠⌋ , (3)

so that expectation values with respect to 𝒵 are dened as

∐𝐴 =∑𝑠𝐴(𝑠⌋ 𝑒

−𝛽𝐻(𝑠⌋

∑𝑠′ 𝑒−𝛽𝐻(𝑠′⌋

. (4)

First we switch o the external magnetic eld ℎ so that the partition function

𝒵 =∑𝑠

𝑒𝛽𝐽∑∐𝑖𝑗 𝑠𝑖𝑠𝑗 , (5)

is a function of 𝛽𝐽 . There are two distinct phases in the model. If 𝛽𝐽 is small, all congurationshave comparable weights (probabilities) in 𝒵, since however strongly 𝐻 uctuates, it is damped bythe small prefactor in the exponent. This means that basically everything is averaged out in 𝒵.This is the disordered phase at high temperature (𝑇 ⇑𝐽 is large). In the opposite limit, 𝛽𝐽 is large,so only the congurations with low values of 𝐻 are relevant in 𝒵. These minimal-energy (maximal-probability) congurations are the ones where the spins are aligned with each other. Since thesystem possesses parity symmetry

𝑃 ∶ 𝑠𝑖 → −𝑠𝑖, ∀𝑖 (6)

4

5

there are two preferred congurations: 𝑠𝑖 = +1 and 𝑠𝑖 = −1. This is the ordered phase at lowtemperature (𝑇 ⇑𝐽 is small).

The two phases can be characterized by the order parameter. In our case this is the magnetiza-tion

∐𝑀 = 𝑇𝜕 log𝒵

𝜕ℎ= ∑

𝑖

𝑠𝑖 , (7)

which is ±1 at 𝑇 = 0 and vanishes at high 𝑇 . At low temperatures ∐𝑀 breaks parity symmetry,since it is odd under the transformation (6). This is spontaneous symmetry breaking: 𝐻 is symmetricbut the ground state of the system is not.

So far we formulated the system in two dimensions, but it turns out that the phase diagramof the Ising model depends crucially on the dimensionality 𝑑 of the system. For 𝑑 = 1, no phasetransition occurs and the system is always in the disordered phase for 𝑇 > 0 → Exercise. In 𝑑 = 2, thecritical temperature 𝑇𝑐 separating the disordered and ordered phases can be found analytically [1],

𝑑 = 2 ∶ 𝑇𝑐 =2𝐽

log(1 +⌋

2)≈ 2.6 ⋅ 𝐽 ,

∐𝑀

𝑁= ± ⌊1 −

1

sinh(2𝐽⇑𝑇 )

1⇑8

⋅Θ(𝑇 − 𝑇𝑐) . (8)

At 𝑇 = 𝑇𝑐 the magnetization is continuous but its derivative 𝜕𝑀⇑𝜕𝑇 has a jump (see Fig. 1.) Thiskind of singularity is a feature of a second-order phase transition. This singularity is only obtainedin the model if the volume is innite. Indeed, if 𝑁 is nite, the number of possible congurations 𝑠 isalso nite, which means that 𝒵, being a nite sum of analytical functions, must also be analytical.The only way a singularity can appear is if the sum is innite. The limit 𝑁 → ∞ is called thethermodynamic limit. The results (8) were obtained in this limit.

Figure 1: Magnetization in the two-dimensional Ising model at zero external eld (green solid line)and at a positive magnetic eld (blue dashed line).

In the lecture we simulated the partition function numerically by a Metropolis algorithm. Start-ing from a random conguration, random spin ips are oered and accepted with probabilitymin1, 𝑒−𝛽Δ𝐻, where ∆𝐻 is the change in the Hamiltonian. The magnetization is indeed observedto show a second-order phase transition (visible already for large but nite systems) at the predictedcritical temperature.

6 PREFACE. ISING MODEL

If the external magnetic eld ℎ is switched on, the symmetry (6) is broken explicitly, and ∐𝑀

is never exactly zero. The phase transition is smeared out and becomes an analytic crossover (seeFig. 1), which exhibits no singularity. (The magnetization cannot be obtained in closed form forℎ > 0; the gure contains a sketch of ∐𝑀.)

Up to now we discussed 𝐽 > 0, i.e. a ferromagnetic interaction. For 𝐽 < 0 we have an an-tiferromagnetic coupling instead that prefers that nearest neighbors point oppositely. For hightemperatures the system is again in the disordered phase, since all congurations have comparableprobabilities. Similarly to 𝐽 > 0, at low 𝑇 the minimal-energy congurations become the groundstate. There are two such congurations, where the spins form a checkerboard: 𝑠𝑖 = +1 on even sitesand 𝑠𝑖 = −1 on odd sites. (A site is called even/odd if 𝑖𝑥+𝑖𝑦 is even/odd.) More concisely, the groundstate is 𝑠𝑖 = (−1)𝑖𝑥+𝑖𝑦 or 𝑠𝑖 = −(−1)𝑖𝑥+𝑖𝑦 , which are again connected by the parity transformation.The order parameter is now the staggered magnetization

𝑀𝑠 =∑𝑖

(−1)𝑖𝑥+𝑖𝑦𝑠𝑖 , (9)

which is ±1 for the two ground states and uctuates around zero for disordered congurations. Wecan write the expectation value of 𝑀𝑠 as a derivative of the partition function if we introduce astaggered magnetic eld ℎ𝑠 in the Hamiltonian,

𝐻(𝑠⌋ = −𝐽 ∑∐𝑖𝑗

𝑠𝑖𝑠𝑗 − ℎ𝑠∑𝑖

(−1)𝑖𝑥+𝑖𝑦𝑠𝑖 , (10)

so that

∐𝑀𝑠 = 𝑇𝜕 log𝒵

𝜕ℎ𝑠. (11)

Again one can prove that in one dimension ∐𝑀𝑠 = 0 for all 𝑇 > 0 and no phase transition occurs →Exercise.

Chapter 1

Quantum mechanics

Let us consider the quantum mechanics of a particle of mass 𝑚, moving in a potential 𝑉 (𝑥). Inparticular, we will specialize to the harmonic oscillator 𝑉 (𝑥) =𝑚𝜔2𝑥2⇑2. The wave function ⋃𝜓 isdetermined (up to a phase) by the time-dependent Schrödinger equation,

𝑖ℎ𝜕

𝜕𝑡⋃𝜓 = ⋃𝜓 , =

𝑝2

2𝑚+ 𝑉 () , (1.1)

where we used natural units 𝑐 = 𝑘𝐵 = 1. The coordinate and momentum operators do not commute:

(, 𝑝⌋ = 𝑖ℎ . (1.2)

The equilibrium value of any observable 𝐴 can be calculated by means of expectation valueswith respect to the partition function 𝒵. At nite temperature 𝑇 it reads

𝒵 = tr 𝑒−𝛽 , ∐𝐴 =1

𝒵tr [𝐴𝑒−𝛽⌉ , (1.3)

where 𝛽 = 1⇑𝑇 .

We will evaluate the trace in 𝒵 for the harmonic oscillator in two dierent bases: in the eigen-modes of (energy basis) and in the coordinate basis. For the former, the basis is spanned by the⋃𝜓𝑛,

⋃𝜓𝑛 = 𝐸𝑛⋃𝜓𝑛, ∐𝜓𝑛⋃𝜓𝑚 = 𝛿𝑛𝑚, 𝐸𝑛 = ℎ𝜔(𝑛 + 1⇑2), 𝑛 ∈ Z+0 . (1.4)

The partition function becomes

𝒵 =∞

∑𝑛=0

∐𝜓𝑛⋃𝑒−𝛽

⋃𝜓𝑛 =∞

∑𝑛=0

𝑒−𝛽𝐸𝑛 = 𝑒−𝛽ℎ𝜔⇑2∞

∑𝑛=0

(𝑒−𝛽ℎ𝜔)𝑛 = 𝑒−𝛽ℎ𝜔⇑21

1 − 𝑒−𝛽ℎ𝜔=

1

2 sinh(𝛽ℎ𝜔⇑2), (1.5)

which, in the classical limit ℎ→ 0, gives (→ Exercise)

𝒵cl =𝑇

ℎ𝜔. (1.6)

From now on, we set ℎ = 1.

In most cases, one does not have access to all the energies. Therefore, it is much more useful towrite down the trace in the coordinate basis. We do this in the following and check that we obtainthe same as above.

7

8 CHAPTER 1. QUANTUM MECHANICS

1.1 Basis states

To represent the trace in coordinate basis, we need to dene eigenstates of and of 𝑝:

⋃𝑥 = 𝑥⋃𝑥, 𝑝⋃𝑥 = −𝑖𝜕

𝜕𝑥⋃𝑥 . (1.7)

We will also make use of the momentum basis

𝑝⋃𝑝 = 𝑝⋃𝑝 (1.8)

so that

𝑝∐𝑥⋃𝑝 = ∐𝑥⋃𝑝⋃𝑝 = −𝑖𝜕

𝜕𝑥∐𝑥⋃𝑝 ⇒ ∐𝑥⋃𝑝 = 𝑒𝑖𝑝𝑥 . (1.9)

The normalization is now such that

∐𝑥⋃𝑥′ = 𝛿(𝑥 − 𝑥′), ∫ d𝑥⋃𝑥∐𝑥⋃ = 1, ∫d𝑝

2𝜋⋃𝑝∐𝑝⋃ = 1 , (1.10)

which can be checked using

∫ d𝑥𝑒−𝑖(𝑝−𝑝′)𝑥

= 2𝜋𝛿(𝑝 − 𝑝′) , (1.11)

so that

1 = ∫ d𝑥⋃𝑥∐𝑥⋃ = ∫ d𝑥∫d𝑝

2𝜋∫

d𝑝′

2𝜋⋃𝑝𝑒−𝑖𝑝𝑥𝑒𝑖𝑝

′𝑥∐𝑝′⋃ = ∫

d𝑝

2𝜋⋃𝑝∐𝑝⋃ . (1.12)

1.2 Path integral

Here we follow [2]. The partition function in the coordinate basis reads

𝒵 = ∫ d𝑥∐𝑥⋃𝑒−𝛽 ⋃𝑥 . (1.13)

Now contains both 𝑝 and , which do not commute with each other. Thus if we want to evaluate

𝑒−𝛽 on ⋃𝑥, we need the Baker-Campbell-Hausdor formula,

𝑒𝐴𝑒𝐵 = 𝑒𝐴+𝐵+12(𝐴,𝐵⌋+... , (1.14)

which contains innitely many terms. However, if 𝜖 ≪ 1 is a small parameter, then we havefrom (1.14)

𝑒−𝜖 = 𝑒−𝜖𝑝2⇑(2𝑚)𝑒−𝜖𝑉 () +𝒪(𝜖2) , (1.15)

so that we can act with 𝑝 to the left and with to the right. In particular, using (1.15) and (1.7-1.9)we can write

∐𝑝𝑖⋃𝑒−𝜖

⋃𝑥𝑖 = 𝑒−𝜖𝑝2𝑖 ⇑(2𝑚)𝑒−𝜖𝑉 (𝑥𝑖)∐𝑝𝑖⋃𝑥𝑖 +𝒪(𝜖

2) = 𝑒−𝜖(𝑝

2𝑖 ⇑(2𝑚)+𝑖𝑝𝑖𝑥𝑖⇑𝜖+𝑉 (𝑥𝑖)+𝒪(𝜖)⌋ . (1.16)

So we know that we have to split into small pieces in order to separate the terms containing𝑝 and those with . So let us separate into 𝑁 pieces and set 𝜖 = 𝛽⇑𝑁 . We also need to insert

1.2. PATH INTEGRAL 9

momentum states ⋃𝑝𝑖 to be able to use (1.16). We insert a full set of momentum states left of 𝑒−𝜖

and a full set of coordinate states to the right of it, and repeat this procedure 𝑁 times.

𝒵 = ∫ d𝑥∐𝑥⋃𝑒−𝜖 . . . 𝑒−𝜖 ⋃𝑥

= ∫ d𝑥∐𝑥⋃∫d𝑝𝑁2𝜋

⋃𝑝𝑁 ∐𝑝𝑁 ⋃𝑒−𝜖∫ d𝑥𝑁 ⋃𝑥𝑁 ∐𝑥𝑁 ⋃∫

d𝑝𝑁−1

2𝜋⋃𝑝𝑁−1∐𝑝𝑁−1⋃𝑒

−𝜖∫ d𝑥𝑁−1⋃𝑥𝑁−1

. . . ∐𝑝1⋃𝑒−𝜖∫ d𝑥1⋃𝑥1∐𝑥1⋃𝑥 .

(1.17)

The last scalar product gives 𝛿(𝑥1 − 𝑥) and integrating over 𝑥 replaces the rst ∐𝑥⋃ by ∐𝑥1⋃. Let usfor the moment denote 𝑥𝑁+1 = 𝑥1. There are altogether 𝑁 integrals over 𝑝𝑖 and 𝑥𝑖. Using (1.16)and (1.9) we obtain

𝒵 =𝑁

∏𝑖=1∫

d𝑥𝑖d𝑝𝑖2𝜋

∐𝑥𝑖+1⋃𝑝𝑖∐𝑝𝑖⋃𝑒−𝜖

⋃𝑥𝑖 =𝑁

∏𝑖=1∫


𝑒𝑖𝑝𝑖𝑥𝑖+1𝑒−𝜖(𝑝2𝑖 ⇑(2𝑚)+𝑖𝑝𝑖𝑥𝑖⇑𝜖+𝑉 (𝑥𝑖)+𝒪(𝜖)⌋ . (1.18)

We can get rid of the 𝒪(𝜖2) terms if we take the limit 𝜖 → 0, which is equivalent to 𝑁 → ∞.Altogether we have

𝒵 = lim𝑁→∞

∫

𝑁

∏𝑖=1


exp

⎨⎝⎝⎝⎝⎪

−𝜖𝑁

∑𝑗=1

⎛

⎝

𝑝2𝑗

2𝑚− 𝑖𝑝𝑗

𝑥𝑗+1 − 𝑥𝑗

𝜖+ 𝑉 (𝑥𝑗)

⎞

⎠

⎬⎠⎠⎠⎠⎮𝑥𝑁+1=𝑥1

. (1.19)

We are left with a Gaussian integral for the momenta that can be evaluated easily


∫

𝑁

∏𝑖=1

d𝑥𝑖⌈

2𝜋𝜖⇑𝑚exp

⎨⎝⎝⎝⎝⎪

−𝜖𝑁

∑𝑗=1

(𝑚

2(𝑥𝑗+1 − 𝑥𝑗

𝜖)2

+ 𝑉 (𝑥𝑗))

⎬⎠⎠⎠⎠⎮𝑥𝑁+1=𝑥1

, (1.20)

which can be recast into a continuum-like-form if we let the continuous variable 𝜏 take the placeof the index 𝑗 (so 𝜖∑𝑗 = ∫ d𝜏), denote the integral over all the (continuously many) 𝑥 by 𝒟𝑥, andputting the (divergent) prefactors into the constant 𝐶:

𝒵 = 𝐶 ∫𝑥(𝛽)=𝑥(0)

𝒟𝑥𝑒−𝑆 , 𝑆 = ∫

𝛽

0d𝜏 ℒ, ℒ =

𝑚

2(𝜕𝑥

𝜕𝜏)

2

+ 𝑉 (𝑥(𝜏)) . (1.21)

The constant 𝐶 is divergent, but it cancels from expectation values (1.3). This expression is verysimilar to the usual, zero-temperature path integral in Minkowski space, which includes

𝑒𝑖𝑆𝑀 , 𝑆𝑀 = ∫

∞

−∞d𝑡ℒ𝑀 , ℒ𝑀 =

𝑚

2(𝜕𝑥

𝜕𝑡)

2

− 𝑉 (𝑥(𝑡)) . (1.22)

In fact, we can obtain the partition function at nite temperature from this expression if we performa Wick-rotation 𝑡 → 𝜏 = 𝑖𝑡 and set ℒ = −ℒ𝑀 . In addition, we also need to restrict 𝜏 to the niteinterval 0 ≤ 𝜏 ≤ 𝛽 and prescribe periodic boundary conditions for the coordinate: 𝑥(𝛽) = 𝑥(0). As itturns out, this prescription works not only in quantum mechanics but also in quantum eld theory.


1.3 The harmonic oscillator

To evaluate 𝒵, we need to specify the potential 𝑉 (𝑥). The simplest example is that of the harmonicoscillator, 𝑉 (𝑥) =𝑚𝜔2𝑥2⇑2. Since ℒ does not depend on 𝑡 explicitly, we can proceed by performinga Fourier-expansion of 𝑥. For a simple treatment of the Fourier expansion (in particular, of theJacobian of the transformation), it is useful to go back to the discretized time variable 𝜏𝑘 = 𝜖𝑘 withthe time interval split in 𝑁 steps. For convenience we take 𝑁 to be odd. The Fourier-transform isa nite sum

𝑥(𝜏𝑘) =1

⌋𝑁∑𝑛

𝑥𝑛 𝑒𝑖𝜔𝑛𝜏𝑘 =

1⌋𝑁∑𝑛

𝑥𝑛 𝑒𝑖2𝜋𝑛𝑘⇑𝑁 , (1.23)

where we exploited the periodicity of 𝑥(𝜏𝑘)

𝑥(𝜏0) = 𝑥(𝜏𝑁) ⇒ 𝜔𝑛𝛽 = 2𝜋𝑛, 𝑛 = 0 . . .𝑁 − 1 , (1.24)

The 𝜔𝑛 are called Matsubara frequencies. Moreover, the reality of the coordinate implies

𝑥(𝜏𝑘)∗= 𝑥(𝜏𝑘) ⇒ 𝑥∗𝑛 = 𝑥𝑁−𝑛 . (1.25)

This means that we can write the Fourier transformation in a more compact form. Denoting𝑥𝑛 = 𝑎𝑛 + 𝑖𝑏𝑛 and noting that (1.25) means that 𝑎𝑁−𝑛 = 𝑎𝑛 and 𝑏𝑁−𝑛 = −𝑏𝑛:

𝑥(𝜏𝑘) =1

⌋𝑁

⎨⎝⎝⎝⎝⎪

𝑎0 + 2(𝑁−1)⇑2

∑𝑛=1

(𝑎𝑛 cos(𝜔𝑛𝜏𝑘) − 𝑏𝑛 sin(𝜔𝑛𝜏𝑘))

⎬⎠⎠⎠⎠⎮

, (1.26)

remember we took 𝑁 to be odd. Here we used that cos(𝜔𝑛𝜏𝑘) = cos(𝜔𝑁−𝑛𝜏𝑘) and sin(𝜔𝑛𝜏𝑘) =

− sin(𝜔𝑁−𝑛𝜏𝑘). The reality of 𝑥(𝜏𝑘) also results in 𝑏0 = 0.The Jacobian of this transformation 𝑥(𝜏𝑘)→ 𝑎𝑛, 𝑏𝑛 is → Exercise

⋃det𝐽 ⋃ = 2(𝑁−1)⇑2 . (1.27)

1.3.1 Gaussian integrals

The normalization factor in the path integral (1.20) seems to diverge as 𝜖 → 0 (or equivalently,𝑁 → ∞) and questions our approach. To see that 𝒵 is nevertheless a nite function, we nowcarefully perform the Gaussian integral over the Fourier-transformed variables.

We insert the Fourier transform in the action. At this point it makes more sense to workwith (1.23) instead of (1.26). Putting the former into the action in (1.20) we get

𝑆 =𝑚𝜖

2∑𝑘

⎨⎝⎝⎝⎝⎪

(𝑥(𝜏𝑘+1) − 𝑥(𝜏𝑘)

𝜖)

2

+ 𝜔2𝑥2(𝜏𝑘)

⎬⎠⎠⎠⎠⎮

=𝑚𝜖

2𝑁∑𝑘

𝑁−1

∑𝑛,𝑚=0

]1

𝜖2(𝑒𝑖𝜔𝑛𝜖 − 1)(𝑒𝑖𝜔𝑚𝜖

− 1) + 𝜔2𝑥𝑛𝑥𝑚𝑒

𝑖2𝜋𝑘(𝑛+𝑚)⇑𝑁

=𝑚𝜖

2

𝑁−1

∑𝑛=0

(4

𝜖2sin2 𝜔𝑛𝜖

2+ 𝜔2

)𝑥𝑛𝑥𝑁−𝑛 =𝑚𝜖

2

𝑁−1

∑𝑛=0

(2𝑛 + 𝜔

2) ⋃𝑥𝑛⋃2 ,

(1.28)

where in the second step we performed the sum over 𝑘, resulting in 𝑁𝛿𝑚,𝑁−𝑛, then summed over𝑚 and used (1.25). In the last step we denoted 𝑛 = 2 sin(𝜔𝑛𝜖⇑2)⇑𝜖. Note that lim𝜖→0 𝑛 = 𝜔𝑛 for𝜔𝑛𝜖 < 𝜋, i.e. for 𝑛 < 𝑁⇑2.

1.4. EQUATION OF STATE 11

Now we can insert 𝑥𝑛 = 𝑎𝑛 + 𝑖𝑏𝑛. Then we have ⋃𝑥𝑛⋃2 = 𝑎2𝑛 + 𝑏

2𝑛 and also 𝑎2𝑁−𝑛 = 𝑎2𝑛, 𝑏

2𝑁−𝑛 = 𝑏2𝑛

and 2𝑁−𝑛 =

2𝑛, which enables us to simplify

𝑆 =𝑚𝜖

2𝜔2𝑎20 +𝑚𝜖

(𝑁−1)⇑2

∑𝑛=1

(2𝑛 + 𝜔

2) (𝑎2𝑛 + 𝑏2𝑛) , (1.29)

and the partition function reads, inserting (1.27)


(2𝜋𝜖⇑𝑚)−𝑁⇑2

⋃det𝐽 ⋃∫ d𝑎0∫⎛

⎝

(𝑁−1)⇑2

∏𝑛=1

d𝑎𝑛d𝑏𝑛⎞

⎠exp (−𝑆⌋

= lim𝑁→∞

(2𝜋𝜖⇑𝑚)−𝑁⇑2

⋅ 2(𝑁−1)⇑2⋅

2𝜋

𝑚𝜖𝜔2⋅

(𝑁−1)⇑2

∏𝑛=1

𝜋

𝑚𝜖(2𝑛 + 𝜔

2)

=𝑇

𝜔lim𝑁→∞

𝑁(𝑁−1)⇑2

∏𝑛=1

⌊4 sin2 𝜋𝑛

𝑁+ (

𝜔

𝑁𝑇)2

−1

,

(1.30)

where we reinserted 𝜖 = 1⇑(𝑁𝑇 ) and 𝑛 and reorganized the product. Notice that factors of 𝑚⇑(2𝜋)canceled from the result. Now we use the identity → Exercise

for 𝑁 odd ∶ 𝑁 =

(𝑁−1)⇑2

∏𝑛=1

4 sin2 𝜋𝑛

𝑁. (1.31)

Thus we can merge 𝑁 with the product to obtain

𝒵 =𝑇

𝜔lim𝑁→∞

(𝑁−1)⇑2

∏𝑛=1

⎨⎝⎝⎝⎝⎪

1 + (𝜔⇑𝑇

2𝑁 sin 𝜋𝑛𝑁

)

2⎬⎠⎠⎠⎠⎮

−1

=𝑇

𝜔

∞

∏𝑛=1

⌊1 + (𝜔

2𝜋𝑛𝑇)2

−1

=1

2⌊𝑧

∞

∏𝑛=1

(1 +𝑧2

𝜋2𝑛2)

−1

𝑧=𝜔⇑(2𝑇 )

=1

2]sinh

𝜔

2𝑇−1

.

(1.32)

Here we expanded the sin for large 𝑁 , so that 𝑁 only appears as the upper limit of the product,where we can send it to innity. This product is just the representation of the sinh function. Indeed,we recovered (1.5) exactly. (Remember we set ℎ = 1 in the meantime.)

A few remarks are in order. The computation of 𝐶 is not really necessary if we are only interestedin expectation values. If we need log𝒵 itself which is the case if we are interested in the freeenergy of the system, as we will see later then we need to keep track of the normalization as well.In our case, 𝒵 is a nite quantity, although this is not transparent throughout the derivation usingthe path integral. For quantum eld theories, it turns out that 𝒵 becomes innite, but this is notdue to the path integral representation but due to ultraviolet divergences in the theory.

Another observation concerns the analyticity of 𝒵: it is singular at 𝜔 = 0. Note that this comesfrom the presence of a zero Matsubara-frequency 𝜔0 = 0. This will have drastic consequences forbosonic eld theories.

1.4 Equation of state

For a general thermodyamical system, the partition function 𝒵 contains information about allequilibrium observables that characterize the system. For the harmonic oscillator, we may denethe free energy 𝐹 , the energy 𝐸 and the entropy 𝑆:

𝐹 = −𝑇 log𝒵, 𝐸 = ∐ =1

𝒵tr (𝑒−𝛽) = −

𝜕 log𝒵

𝜕𝛽, (1.33)


𝑆 = −𝜕𝐹

𝜕𝑇= log𝒵 + 𝑇

1

𝒵tr (𝑒−𝛽⇑𝑇 2

) = −𝐹

𝑇+𝐸

𝑇. (1.34)

Let's calculate these quantities for the harmonic oscillator. The free energy reads

𝐹 = 𝑇 log [𝑒𝜔⇑(2𝑇 ) − 𝑒−𝜔⇑(2𝑇 )⌉ = 𝑇 log [𝑒𝜔⇑(2𝑇 ) (1 − 𝑒−𝜔⇑𝑇 )⌉ =𝜔

2)])𝐹vac

+𝑇 log(1 − 𝑒−𝜔⇑𝑇 ))⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊]⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊)

𝐹therm

. (1.35)

Notice that there is a temperature-independent term that is present even at 𝑇 = 0. This is thevacuum free energy. The rest is the thermal free energy. The entropy is the derivative of this withrespect to 𝑇 , so it has no vacuum term:

𝑆 = − log(1 − 𝑒−𝜔⇑𝑇 ) + 𝑇𝑒−𝜔⇑𝑇

1 − 𝑒−𝜔⇑𝑇𝜔

𝑇 2= − log(1 − 𝑒−𝜔⇑𝑇 ) +

𝜔

𝑇

1

𝑒𝜔⇑𝑇 − 1(1.36)

thus it vanishes at 𝑇 = 0. We might interpret this by saying that the zero-point energy of theoscillator has no physical particle interpretation. In quantum eld theory we will see that thisenergy is related to virtual particles. The energy is

𝐸 = 𝐹 + 𝑇𝑆 =𝜔

2+

𝜔

𝑒𝜔⇑𝑇 − 1. (1.37)

1.5 Summary for the harmonic oscillator

For later reference, we express the partition function of the harmonic oscillator in several dierentforms. We take (1.30) and perform the limit 𝑁 →∞ already in the action. Then we lose track ofthe normalization constant, but this cancels in expectation values anyway. Moreover, we use (1.32)

𝒵 = 𝐶 ′⋅ ∫ 𝒟𝑥 exp ⌊−

𝑚𝜖

2∑𝑛

(𝜔2𝑛 + 𝜔

2)⋃𝑥𝑛⋃

2

=𝑇

𝜔∏𝑛≥1

𝜔2𝑛

𝜔2𝑛 + 𝜔

2

=∞

∏𝑛=−∞

(𝜔2𝑛 + 𝜔

2

𝑇 2)

−1⇑2

𝐶 ′′

(1.38)

where in the last step we rearranged the product and 𝐶 ′′ is an 𝜔-independent constant. Thus, thefree energy reads

𝐹 =𝑇

2

∞

∑𝑛=−∞

log𝜔2𝑛 + 𝜔

2

𝑇 2+ (𝜔-independent⌋

=𝜔

2+ 𝑇 log(1 − 𝑒−𝜔⇑𝑇 )

(1.39)

where we also compared to the form (1.35).

Chapter 2

Real scalar fields

A quantum eld generalizes the concept of the degree of freedom from one coordinate 𝑥(𝑡) to a eld,which is a function of both space and time 𝜑(𝑡,x). The excitations of the eld correspond to theparticles of the theory. The elementary particles of nature all are excitations of various quantumelds. We start with scalar eld theory that describes bosonic particles.

2.1 Path integral

For the harmonic oscillator, we started from the Hamiltonian to write down 𝒵. In eld theories, itis more convenient to consider the Lagrangian formulation (for example it exhibits explicit Lorentzinvariance). To be specic, we take a real scalar quantum eld 𝜑(𝑥) = 𝜑(𝑡,x) in 3+1 space-timedimensions. In Minkowski space (with metric 𝑔𝜇𝜈 = diag(1,−1,−1,−1)), the action is

𝑆𝑀 = ∫ d3x∫ d𝑡ℒ𝑀 , ℒ𝑀 =1

2𝜕𝜇𝜑𝜕

𝜇𝜑 − 𝑉 (𝜑) =1

2(𝜕𝑡𝜑)

2−

1

2(𝜕𝑖𝜑)

2− 𝑉 (𝜑) . (2.1)

To go over to the Hamiltonian density ℋ, we need to introduce the momenta

𝜋 =𝜕ℒ𝑀𝜕(𝜕𝑡𝜑)

= 𝜕𝑡𝜑 , (2.2)

and do a Legendre transformation

ℋ = 𝜕𝑡𝜑 ⋅ 𝜋 −ℒ =𝜋2

2+

1

2(𝜕𝑖𝜑)

2+ 𝑉 (𝜑) . (2.3)

Comparing this to the canonical momentum and the Hamiltonian of the harmonic oscillator,

𝑝 =𝑚𝜕𝑡𝑥, 𝐻 =𝑝2

2𝑚+ 𝑉 (𝑥) , (2.4)

reveals that the eld theory is nothing else but lots of 𝑚 = 1 harmonic oscillators, located at allthe points x of space and attached to each other via `nearest-neighbor' couplings (𝜕𝑖𝜑)

2⇑2. Theequivalent of the momentum 𝑝 becomes the canonical momentum 𝜋.

We want to derive the analogue of (1.21) for 𝒵. To arrive at that formula, we needed to do aGaussian integration in the momenta 𝑝. It was essential that the Hamiltonian is quadratic in 𝑝.Also here the Hamiltonian is quadratic in 𝜋. So the formula carries over and we have

𝒵 = ∫𝜑(𝛽,x)=𝜑(0,x)

∏x

(𝐶 ⋅𝒟𝜑(𝜏,x)) 𝑒−𝑆 , 𝑆 = ∫

𝛽

0d𝜏 ∫ d3xℒ , (2.5)

13

14 CHAPTER 2. REAL SCALAR FIELDS

with the Euclidean Lagrangian

ℒ = −ℒ𝑀(𝑡→ 𝜏 = 𝑖𝑡) =1

2𝜕𝜇𝜑𝜕𝜇𝜑 + 𝑉 (𝜑) =

1

2(𝜕𝜏𝜑)

2+

1

2(𝜕𝑖𝜑)

2+ 𝑉 (𝜑) , (2.6)

where both Lorentz-indices are subscripts to denote the Euclidean metric.

To evaluate the path integral, we specify the potential 𝑉 = 𝑚2𝜑2⇑2 which corresponds to thefree case. The next step just as was for the harmonic oscillator is to go over to Fourier-space.For the harmonic oscillator, we had to be careful and retain the discretized time in order to keeptrack of the normalization constant. Here we simplify things by letting time be continuous. Sincenow we have an additional continuous variable x, it will be useful to restrict the system to a nitevolume 𝑉 = 𝐿1𝐿2𝐿3, impose periodic boundary conditions and perform a Fourier-transformationin this (also continuous) variable as well,

𝜑(𝜏,x) = 𝑇∑𝑛

1

𝑉∑k

𝜑(𝜔𝑛,k) 𝑒𝑖𝜔𝑛𝜏+𝑖kx . (2.7)

The periodicity conditions imply

𝜔𝑛 = 2𝜋𝑛𝑇, 𝑛 ∈ Z, 𝑘𝑖 =2𝜋𝑛𝑖𝐿𝑖

, 𝑛𝑖 ∈ Z . (2.8)

The spatial boundary conditions will not matter in the large volume limit where we have

1

𝐿𝑖∑𝑘𝑖

𝐿→∞ÐÐÐ→ ∫

d𝑘𝑖2𝜋

. (2.9)

In addition, the reality of the eld implies

𝜑(𝜔𝑛,k)∗= 𝜑(−𝜔𝑛,−k) . (2.10)

Putting the Fourier-transformed eld in the action, we follow the same lines as for the harmonicoscillator. For example the second term in the action becomes

∫

𝛽

0d𝜏 ∫ d3x

(𝜕𝑖𝜑)2

2= ∫

𝛽

0d𝜏 ∫ d3x

𝑇 2

2𝑉 2 ∑𝑛,𝑚∑k,p

𝜑(𝜔𝑛,k)𝜑(𝜔𝑚,p)(𝑖𝑘𝑖)(𝑖𝑝𝑖)𝑒𝑖(𝜔𝑛+𝜔𝑚)𝜏𝑒𝑖(p+k)x

=𝑇

2𝑉∑𝑛∑k

⋃𝜑(𝜔𝑛,k)⋃2𝑘2𝑖 ,

(2.11)

where we rst integrated over 𝜏 and x resulting in 𝑉 ⇑𝑇 ⋅ 𝛿𝑚,−𝑛𝛿p,−k, then summed over 𝑚 and p,and nally exploited (2.8) and (2.10). Similarly, the full action then reads

𝑒−𝑆 = exp ⌊−𝑇

2𝑉∑𝑛∑k

(𝜔2𝑛 + k2

+𝑚2)⋃𝜑(𝜔𝑛,k)⋃

2 =∏

k

exp ⌊−𝑇

2𝑉∑𝑛

(𝜔2𝑛 + k2

+𝑚2)⋃𝜑(𝜔𝑛,k)⋃

2 .

(2.12)The comparison to (1.28) reveals that path integral factorizes into a harmonic-oscillator-type pathintegral for each k, if we assign

𝑥𝑛 ↔ 𝜑(𝜔𝑛,k), 𝜔↔⌋k2 +𝑚2 ≡ 𝐸k . (2.13)

2.2. PROPER TIME REPRESENTATION 15

So we do not need to perform the calculation, we can just read it o from our earlier results,summarized in various forms in (1.38). Remember that the prefactor in the exponent drops out inthe nal formulae for 𝒵.

𝒵 =∏𝑘

∏𝑛

(𝜔2𝑛 +𝐸

2k)

−1⇑2⋅𝐶 ′′ . (2.14)

Comparing to (1.39) we see that the free energy density of the real scalar eld is

𝑓 =𝐹

𝑉= ∫

d3k

(2𝜋)3]𝐸k

2+ 𝑇 log(1 − 𝑒−𝐸k⇑𝑇 ) (2.15)

= ∫d3k

(2𝜋)3𝑇

2

∞

∑𝑛=−∞

log𝜔2𝑛 +𝐸

2k

𝑇 2+ (𝐸k-independent⌋ (2.16)

where we went to the large volume limit and used (2.9). It can be checked that the two expressionsare equal → Exercise.

Note that (2.13) dictates that the real scalar eld inherits the non-analiticity of the harmonicoscillator. While for the latter 𝒵 had a singularity at 𝜔 = 0, in the present case this translates tozero energies, i.e. 𝑚 = 0. We will get back to this in below.

Just as 𝐹 for the harmonic oscillator, the free energy of real scalar elds also separates into avacuum term and a thermal contribution. The former is apparently divergent (it is the integral of𝐸k ≈ 𝑘 for large 𝑘), whereas the latter is nite (the integrand decays exponentially for large 𝑘).The divergence of the vacuum term is unphysical: it cancels in dierences of free energies. Still, thevacuum term does give rise to physical eects. One example is them Casimir-eect, which arisesif the momenta in the vacuum term are constrained by boundary conditions (force between twoplates of a capacitor). Later we will see another example when the vacuum term is aected bybackground electromagnetic elds.

One more remark to make: the relation between (2.12) and (2.14) can be carried over to a moreabstract level. Rewriting the exponent in matrix notation (we are sloppy here and ignore the factthat 𝜑 has a dimension)

𝑒−𝑆 = exp )−𝜑𝐷𝜑 ⌈ (2.17)

shows that the path integral is actually → Exercise

𝒵 = 𝐶 ′′⋅ (det𝐷)−1⇑2 . (2.18)

2.2 Proper time representation

We start from the second representation of the free energy density of the real scalar eld:

𝑓 =1

2⨋𝑛,k

log𝜔2𝑛 +𝐸

2k

𝑇 2, ⨋

𝑛,k= ∫

d3k

(2𝜋)3𝑇∑

𝑛

, 𝐸2k = k2

+𝑚2 . (2.19)

Clearly, this is a divergent integral. We need to regularize it in order to have it under control. Thiscan be done in the most transparent manner if we work out a third representation for 𝑓 . To thisend we will use the Mellin transform (A.18) of the argument 𝜆 of the logarithm

(𝜆2)−𝛼 =1

Γ(𝛼)∫

∞

0d𝑠 𝑠𝛼−1 𝑒−𝜆

2𝑠, Re 𝜆2 > 0 , (2.20)

so that by rewriting the logarithm we obtain

log𝜆2 = (𝜆2)−𝛼 log𝜆2⋂𝛼=0

= −𝜕

𝜕𝛼(𝜆2)−𝛼⋂

𝛼=0= −

𝜕

𝜕𝛼⋀𝛼=0

1

Γ(𝛼)∫

∞

0d𝑠 𝑠𝛼−1 𝑒−𝜆

2𝑠 . (2.21)


Putting this together, the Mellin transform for the free energy density gives

𝑓 = −1

2

𝜕

𝜕𝛼⋀𝛼=0

1

Γ(𝛼)⨋𝑛,k∫

∞

0d𝑠 𝑠𝛼−1 𝑒−(𝜔

2𝑛+k

2+𝑚2)𝑠⇑𝑇 2

. (2.22)

Note that all ènergies' are strictly positive if 𝑚 ≠ 0 so that the Mellin transform is well-dened.Now we can interchange the integral over 𝑠 with the sum over Matsubara frequencies and theintegral over momenta, and perform them:

∫

∞

−∞d𝑘𝑖 𝑒

−𝑘2𝑖 𝑠⇑𝑇2

= 𝑇⌈𝜋⇑𝑠,

∞

∑𝑛=−∞

𝑒−𝜔2𝑛𝑠⇑𝑇

2

=1

2⌋𝜋𝑠

Θ3 (0, 𝑒−1⇑(4𝑠)) , (2.23)

where Θ3 is an elliptic theta function. The dierentiation with respect to 𝛼 can also be done,

𝜕

𝜕𝛼⋀𝛼=0

𝑠𝛼

Γ(𝛼)= 1 , (2.24)

which gets rid of all the 𝛼-dependence. Inserting all this into 𝑓 , and making a change of variables𝑠→ 𝑠𝑇 2, we obtain

𝑓 = −1

32𝜋2∫

∞

0d𝑠

1

𝑠3𝑒−𝑚

2𝑠 Θ3 (0, 𝑒−1⇑(4𝑠𝑇2)) . (2.25)

The zero-temperature limit1 of the elliptic theta function is unity, so

𝑓(𝑇 = 0) = −1

32𝜋2∫

∞

0d𝑠

1

𝑠3𝑒−𝑚

2𝑠 . (2.26)

For later use we can also calculate the free energy density coming solely from the zero Matsubara-mode 𝑛 = 0. We just need to remove the sum over 𝑛 in (2.22) and replace 𝜔𝑛 by 𝜔0 = 0. The resultreads

𝑓𝑛=0 = −𝑇⌋𝜋

16𝜋2∫

∞

0d𝑠

1

𝑠5⇑2𝑒−𝑚

2𝑠 . (2.27)

2.3 Regularization

In each of the three representations that we have derived (2.15, 2.16, 2.25), we have seen that 𝑓 isdivergent. From (2.15) it is clear that the divergence comes from the zero-temperature part only(the 𝑇 -dependent integrand goes to zero exponentially with 𝑘). So we will concentrate on 𝑓(𝑇 = 0).In order to make sense of this quantity, we have to perform some kind of a renormalization. Thiswill not be the usual renormalization that one encounters when dealing with loop diagrams in aeld theory. While the latter is taken care of by a multiplicative redenition of the parameters ofthe theory (coupling, mass, etc.), here we will need an additive renormalization.

To perform the renormalization, the rst step is the regularization of the divergent integral. Wecan do this in various dierent ways: with a cuto, with dimensional regularization or with theso-called 𝜁-function regularization → Exercise.

Usually cuto regularization means setting ⋃k⋃ < Λ with Λ some ultraviolet energy scale. Forthe representation (2.25), the parameter 𝑠 has dimension inverse mass squared. Thus, ultravioletphysics (high energies) is encoded in small 𝑠, whereas the infrared region is at high 𝑠. Implementinga cuto in this case thus means

𝑓(𝑇 = 0) = −1

32𝜋2∫

∞

Λ−2

d𝑠

𝑠3𝑒−𝑚

2𝑠= 𝒪(Λ4

) +𝒪(Λ2) +𝒪(log Λ) . (2.28)

1Whenever 𝑇 = 0 is mentioned it is understood as the limit 𝑇 → 0. Already (2.19) seems ill-defined at 𝑇 = 0, butlim𝑇→0 𝑓(𝑇 ) exists if an UV regularization is performed (see below).

2.4. RENORMALIZATION 17

Another regularization technique that we will use is called 𝜁-function regularization: this amountsto damping the UV divergence by keeping the power 𝛼 > 0 in the expression, cf. (2.22):

𝑓(𝑇 = 0) = −1

32𝜋2𝜇𝛼∫

∞

0

d𝑠

𝑠3−𝛼⇑2𝑒−𝑚

2𝑠= 𝒪(𝛼−1) +𝒪(𝛼0

) . (2.29)

Here we needed to specify a dimensionful parameter 𝜇 to x the dimension of 𝑓 to four. Note that𝜁-function regularization (similarly to dimensional regularization) misses all power-like divergences.This is expected, since this regularization involves no dimensionful scale (the scale 𝜇 can only appearin a logarithm), thus 𝑓 ∝𝑚4 must be fullled.

2.4 Renormalization

The above implies that the quantity 𝑓(𝑇 = 0) is not well-dened. It is not a physical observable,similar to the potential of the electromagnetic eld at a single point. A meaningful quantity is thedierence of potentials at two separated points, in which the undened additive constant cancels.The analogue of this in our case is the dierence 𝑓(𝑇 ) − 𝑓(0). In the two representations (2.15)and (2.25) it is given by

𝑓 therm(𝑇 ) ≡ 𝑓(𝑇 ) − 𝑓(0) = 𝑇 ∫d3k

(2𝜋)3log(1 − 𝑒−𝐸k⇑𝑇 ) (2.30)

= −1

32𝜋2∫

∞

0

d𝑠

𝑠3𝑒−𝑚

2𝑠[Θ3 (0, 𝑒−1⇑(4𝑠𝑇

2)) − 1⌉ . (2.31)

In both cases the integrand falls o exponentially in the problematic part of the integration domain(large 𝑘 or small 𝑠). Thus 𝑓 therm is a renormalized, nite observable. For the representation (2.16)this combination is the dierence of the sum and the integral over the Matsubara frequencies (in the𝑇 → 0 limit the frequencies become innitesimally close to each other). This is not too convenient totreat, so we will work with the above two variants. The rst one will be handy for high-temperatureexpansions, whereas the second is more compact and allows for a simple one-dimensional numericalintegration.

Notice that 𝑓 therm is ill-dened at 𝑚 = 0. This is the most visible in (2.31), where the high-𝑠region of the integral diverges if 𝑚 = 0. This is an infrared divergence to which we get back to whenwe consider interactions.

For later use we return to (2.27) and regularize it with 𝜁-function regularization,

𝑓𝑛=0(𝑇 ) = −𝑇⌋𝜋

16𝜋2𝜇𝛼∫

∞

0

d𝑠

𝑠5⇑2−𝛼⇑2𝑒−𝑚

2𝑠= −

𝑇 ⋃𝑚⋃3

12𝜋+𝒪(𝛼) . (2.32)

2.5 High-temperature expansion

Next we want to look at the thermal free energy at high temperatures. This will reproduce theStefan-Boltzmann limit of an ideal gas and subleading corrections to it. It will also highlight aninteresting non-analiticity of the free energy that is peculiar to scalar elds. For this calculation wewill use what we learned about the Mellin transform.

We start from the thermal part (2.30) of the free energy density. The integral cannot be obtainedin a closed form so we rewrite 𝑓 therm in an expansion that will be useful for an approximation athigh temperatures. Following [3], we Taylor-expand the logarithm

log(1 − 𝑧) = −∞

∑𝑙=1

𝑧𝑙

𝑙, (2.33)


which gives a sum over powers of 𝑒−𝛽𝐸k with 𝛽 = 1⇑𝑇 . Now we will use the Mellin transform `backand forth'. We will have to watch out that the transformation is well-dened. The exponentialsare rst rewritten using the inverse Mellin transform (A.17):

𝑓 therm = −𝑇∞

∑𝑙=1

1

𝑙∫

d3k

(2𝜋)3𝑒−𝛽

⌋k2+𝑚2𝑙

= −𝑇∞

∑𝑙=1

1

𝑙∫

d3k

(2𝜋)31

2𝜋𝑖∫

𝑐+𝑖∞

𝑐−𝑖∞d𝑧 Γ(𝑧)𝑙−𝑧𝛽−𝑧(k2

+𝑚2)−𝑧⇑2 .

(2.34)Now the last factor is inserted into the Mellin transform (valid for Re𝑧 > 0),

(k2+𝑚2

)−𝑧⇑2

=1

Γ(𝑧⇑2)∫

∞

0d𝑡 𝑡𝑧⇑2−1 𝑒−(k

2+𝑚2)𝑡 , (2.35)

and the Gaussian integral over k is done to arrive at

𝑓 therm = −𝑇

8𝜋31

2𝜋𝑖∫

𝑐+𝑖∞

𝑐−𝑖∞d𝑧

∞

∑𝑙=1

𝑙−1−𝑧Γ(𝑧)

Γ(𝑧⇑2)𝛽−𝑧 ∫

∞

0d𝑡 [

⌈𝜋⇑𝑡⌉

3𝑡𝑧⇑2−1 𝑒−𝑚

2𝑡 . (2.36)

The 𝑡-integral gives another Γ-function, while the sum over 𝑙 the 𝜁-function:

∫ d𝑡 𝑡(𝑧−3)⇑2−1 𝑒−𝑚2𝑡= Γ(

𝑧 − 3

2) ⋃𝑚⋃

3−𝑧,∞

∑𝑙=1

𝑙−(1+𝑧) = 𝜁(1 + 𝑧) . (2.37)

Note that the rst relation only holds for Re𝑧 > 3. In addition, the absolute value of the massappears here, since (𝑚2)(3−𝑧)⇑2 = ⋃𝑚⋃3−𝑧. Altogether this gives

𝑓 therm = −𝑇⋃𝑚⋃3

8𝜋⌋𝜋

1

2𝜋𝑖∫

𝑐+𝑖∞

𝑐−𝑖∞d𝑧

Γ(𝑧)Γ((𝑧 − 3)⇑2

Γ(𝑧⇑2)𝛽−𝑧 𝜁(1 + 𝑧) ⋃𝑚⋃

−𝑧 . (2.38)

We can simplify this expression using the duplication formula (A.5)

𝑓 therm = −𝑇⋃𝑚⋃3

16𝜋21

2𝜋𝑖∫

𝑐+𝑖∞

𝑐−𝑖∞d𝑧 Γ(

𝑧 + 1

2)Γ(

𝑧 − 3

2) 𝜁(1 + 𝑧) (

𝛽⋃𝑚⋃

2)

−𝑧

. (2.39)

The strongest condition for the validity of the Mellin transforms was Re𝑧 > 3 so the integrationcontour must have 𝑐 > 3. The remaining integration we will perform using Cauchy's theorem (A.19).To this end we need to identify where we can close the integration contour and we need to locatethe poles of the integrand.

Performing the contour integral and expanding the result in 1⇑𝑇 , we obtain → Exercise

𝑓 therm = −𝜋2

90𝑇 4

+1

24𝑚2𝑇 2

−1

12𝜋⋃𝑚⋃

3𝑇 −1

64𝜋2(2𝛾𝐸 −

3

2+ log

𝑚2

16𝜋2𝑇 2)𝑚4

+𝒪(𝑇−2) . (2.40)

Let us look at the analyticity of 𝑓 therm. We started with the expression (2.30), which only dependson 𝑚2. However, our nal result contains ⋃𝑚⋃3, which is non-analytic around 𝑚 = 0. Where doesthis non-analyticity come from?

When we perform the high-temperature expansion, the dimensionless expansion parameter isactually 𝑚⇑𝑇 . Thus, high temperatures correspond to near-massless elds. The massless limitof scalar eld theory is indeed ill-dened due to the presence of the zero Matsubara-mode as wediscussed at the end of Sec. 2.1. More specically, the integrand of (2.30) for zero momenta behavesas

log (1 − 𝑒−⋃𝑚⋃⇑𝑇) = log(⋃𝑚⋃⇑𝑇 ) +𝒪(⋃𝑚⋃) , (2.41)

and is thus non-analytic in 𝑚. Eventually, this gives rise to the odd power 3 in the expansion (2.40).If we compare the result more closely to (2.32), we can see that the ⋃𝑚⋃3 term is actually exactly

coming from the lowest Matsubara-mode 𝑛 = 0. This is the mode that has the infrared problemsand, thus, causes the non-analyticity.



Using the relations of Sec. 1.4 we can derive all thermodynamic observables from (2.40). Besidesthe extensive quantities that we already introduced for the harmonic oscillator, we also dene thecorresponding densities

𝑓 =𝐹

𝑉, 𝑠 =

𝑆

𝑉, 𝜖 =

𝐸

𝑉. (2.42)

Another quantity relevant for the equation of state is the pressure,

𝑝 = −𝜕𝐹

𝜕𝑉

𝑉→∞ÐÐÐ→ −

𝐹

𝑉= −𝑓 . (2.43)

In a homogeneous system, the free energy (being an extensive quantity) is proportional to thevolume 𝑉 if the system is large enough. Then, the derivative with respect to 𝑉 simplies to adivision by 𝑉 . We also dene the interaction measure (or trace anomaly), which measures thedeviation of the equation of state from ideality 𝜖 = 3𝑝,

𝐼 = 𝜖 − 3𝑝 . (2.44)

Looking at the high-temperature limit of (2.40), we can recover the Stefan-Boltzmann law forthe pressure of real scalar particles

lim𝑇→∞

𝑝therm

𝑇 4= − lim

𝑇→∞

𝑓 therm

𝑇 4=𝜋2

90. (2.45)

The interaction measure reads

𝐼therm =1

12𝑚2𝑇 2

−1

4𝜋⋃𝑚⋃

3𝑇 +1

16𝜋2𝑚4

(1 − 2𝛾𝐸 − log𝑚2

16𝜋2𝑇 2) +𝒪(𝑇−2) . (2.46)

Notice that for massless particles 𝐼therm = 0 signaling that the system is ideal (it consists of masslessnon-interacting particles).


Figure 2.1: High-temperature expansion of the pressure of free scalar particles compared to the fullresult.

Chapter 3

Complex scalar fields

Up to now we discussed the non-interacting scalar eld. Since there is no interaction, the numberof particles is conserved in this theory. However, the presence of interactions would spoil thisand no conserved particle number could be dened. This is due to the fact that there is nocontinuous symmetry in the Lagrangian which, as a consequence of Noether's theorem, would leadto a conserved current 𝑗𝜇. There is only the discrete symmetry 𝜑 → −𝜑. The simplest theory witha continuous symmetry is the complex scalar field.

The Minkowskian Lagrangian for a complex scalar eld 𝜑 reads

ℒ𝑀 = 𝜕𝜇𝜑∗𝜕𝜇𝜑 − 𝑉 (𝜑∗𝜑) , (3.1)

with the continuous symmetry𝜑→ 𝑒𝑖𝛼𝜑 𝜑∗ → 𝑒−𝑖𝛼𝜑∗ . (3.2)

Noether's theorem tells us that the conserved current is

𝑗𝜇 =𝜕ℒ

𝜕(𝜕𝜇𝜑)

𝛿𝜑

𝛿𝛼⋀𝛼=0

+𝜕ℒ

𝜕(𝜕𝜇𝜑∗)

𝛿𝜑∗

𝛿𝛼⋀𝛼=0

= 𝜕𝜇𝜑∗(𝑖𝜑) + 𝜕𝜇𝜑(−𝑖𝜑

∗) = 2 Im(𝜑∗𝜕𝜇𝜑⌋ , (3.3)

and the conserved charge is

𝑁 = ∫ d3x 𝑗0 = 2∫ d3x Im(𝜑∗𝜕0𝜑⌋ . (3.4)

3.1 Grand canonical ensemble

We can either work at a xed particle number 𝑁 , i.e. we remain in the canonical ensemble, orwe can let the particle number uctuate. The conjugate quantity playing a similar role as thetemperature does for the energy uctuations is the chemical potential 𝜇,

𝜇 =𝜕𝐹

𝜕𝑁, (3.5)

or, after a Legendre transformation 𝐹 (𝑇,𝑁)→ Ω(𝑇,𝜇)

Ω = 𝐹 − 𝜇𝑁, 𝑁 = −𝜕Ω

𝜕𝜇, (3.6)

which is called grand potential. We'll also need the number density,

𝑛 =𝑁

𝑉. (3.7)

21

22 CHAPTER 3. COMPLEX SCALAR FIELDS

In a quantum system we work with operators. We dene the grand canonical partition functionvia

Ω = −𝑇 log𝒵GC, 𝒵GC = tr 𝑒−𝛽(−𝜇) , (3.8)

which indeed fullls that the 𝜇-derivative of −Ω gives the expectation value of the particle numberoperator. Now the derivative with respect to 𝛽 is a combination of the energy and the particlenumber,

−𝜕 log𝒵GC

𝜕𝛽= ∐ − 𝜇 = 𝐸 − 𝜇𝑁 . (3.9)

3.2 Path integral

So besides the Hamiltonian, we have the term 𝜇𝑁 in the exponent. Notice that the charge 𝑁contains the canonical momentum (𝜋 = 𝜕0𝜑) conjugate to the eld 𝜑. Thus, it will change thederivation of the path integral slightly. The exponent is still at most quadratic in the momenta,but now a linear piece is also present. Thus it is a shifted Gaussian integral. To calculate it, it isbetter to work with real variables, so we parameterize the eld as

𝜑 =1⌋

2(𝜑1 + 𝑖𝜑2), 𝜋1 = 𝜕0𝜑1, 𝜋2 = 𝜕0𝜑2 . (3.10)

So the new term in the exponent is

−𝜇𝑁 = −𝜇∫ d3x Im((𝜑1 − 𝑖𝜑2)(𝜕0𝜑1 + 𝑖𝜕0𝜑2)⌋ = ∫ d3x (𝜇𝜑2𝜋1 − 𝜇𝜑1𝜋2⌋ . (3.11)

This term enters the exponent of the path integral. Remember we discretized the 𝜏 interval ininnitesimal steps, inserted full sets of momentum and coordinate states and arrived at (1.19). Theanalogue of that expression, neglecting the normalization constant, is

∫ d𝜑1d𝜑2∫ d𝜋1d𝜋2 exp ⌊−∫𝛽

0d𝜏 ∫ d3x (

𝜋212− 𝑖𝜋1𝜕𝜏𝜑1 +

𝜋222− 𝑖𝜋2𝜕𝜏𝜑2 + . . . + 𝜇𝜑2𝜋1 − 𝜇𝜑1𝜋2) ,

(3.12)where the dots denote 𝜋-independent terms. The 𝑖𝜋𝑗𝜕𝜏𝜑𝑗 terms stem from the analogue of the ∐𝑝⋃𝑥terms for the harmonic oscillator. Performing the shifted Gaussian integrals,

∫ d𝜋1 exp ⌊−(𝜋212− 𝑖𝜋1(𝜕𝜏𝜑1 + 𝑖𝜇𝜑2)) = 𝐶 ⋅ exp ]−

1

2(𝜕𝜏𝜑1 + 𝑖𝜇𝜑2)

2 , (3.13)

and, similarly, with a minus sign in front of the chemical potential for the momenta 𝜋2. Going backto the complex elds this amounts to

1

2(𝜕𝜏𝜑1 + 𝑖𝜇𝜑2)

2+

1

2(𝜕𝜏𝜑2 − 𝑖𝜇𝜑1)

2=

1

2)(𝜕𝜏𝜑1)

2+ (𝜕𝜏𝜑2)

2− 𝜇2𝜑21 − 𝜇

2𝜑22⌈ + 𝑖𝜇(𝜑2𝜕𝜏𝜑1 − 𝜑1𝜕𝜏𝜑2)

= 𝜕𝜏𝜑∗𝜕𝜏𝜑 − 𝜇

2𝜑∗𝜑 + 𝜇(𝜑𝜕𝜏𝜑∗− 𝜑∗𝜕𝜏𝜑) = (𝜕𝜏 − 𝜇)𝜑

∗(𝜕𝜏 + 𝜇)𝜑 ,

(3.14)

showing that the usual derivative has to be replaced by the covariant derivative 𝐷𝜏 = 𝜕𝜏 ±𝜇, the signdepending on whether it acts on the eld 𝜑 or on the conjugate eld 𝜑∗. The covariant derivativein the spatial directions remains the ordinary derivative. Altogether the path integral reads

𝒵GC = 𝐶 ⋅ ∫ 𝒟𝜑 exp ]−∫𝛽

0d𝜏 ∫ d3x ((𝜕𝜏 − 𝜇)𝜑

∗(𝜕𝜏 + 𝜇)𝜑 + 𝜕𝑖𝜑

∗𝜕𝑖𝜑 + 𝑉 (𝜑∗𝜑)) , (3.15)

3.2. PATH INTEGRAL 23

and in the following we again consider the free case 𝑉 (𝜑∗𝜑) =𝑚2𝜑∗𝜑.

The next step is to go to Fourier space which introduces the Matsubara frequencies. For therst term in the exponent of (3.15) we have

(−𝑖𝜔𝑛 − 𝜇)(𝑖𝜔𝑛 + 𝜇) = (𝜔𝑛 − 𝑖𝜇)2 , (3.16)

so that the equivalent of (2.12) becomes

exp(−𝑆⌋ =∏k

exp ⌊ −𝑇

𝑉∑𝑛

((𝜔𝑛 − 𝑖𝜇)2+ k2

+𝑚2

)⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊]⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊)𝐸2

k

)⋃𝜑(𝜔𝑛,k)⋃2 . (3.17)

This means that two things have changed. First, since we have no reality constraint, all components𝑛 and k are independent and are integrated over. Second, the Matsubara frequencies are shifted inthe imaginary direction by the chemical potential.

The Gaussian integrals over the complex 𝜑 can be performed and give almost the same as wehad in the real case. Comparing to Eq. (2.18) we now have → Exercise

𝑒−𝑆 = exp )−𝜑∗𝐷𝜑 ⌈ → 𝒵GC = 𝐶 ′⋅ (det𝐷)−1 . (3.18)

Comparing this expression to (2.16) we read o that the grand potential Ω = −𝑇 log𝒵GC is

Ω

𝑉= ∫

d3k

(2𝜋)3𝑇

∞

∑𝑛=−∞

log(𝜔𝑛 − 𝑖𝜇)

2 +𝐸2k

𝑇 2+ (𝐸k-independent⌋ . (3.19)

However, we will see below in Sec. 3.3 that this formula becomes ill-dened at 𝜇 = 𝑚. Con-straining the discussion for the moment to 𝜇 < 𝑚, we can recast this in a form similar to (2.15)using the same trick as for the real scalar eld: dierentiating with respect to the energy,

𝑇𝜕

𝜕𝐸

∞

∑𝑛=−∞


2 +𝐸2

𝑇 2=∑

𝑛

𝐸⇑(2𝑇 )

(𝑛𝜋 − 𝑖𝜇⇑(2𝑇 ))2 + (𝐸⇑(2𝑇 ))2=

1

2]coth

𝐸 − 𝜇

2𝑇+ coth

𝐸 + 𝜇

2𝑇 ,

(3.20)and then integrating back (leaving aside 𝐸-independent terms, denoted by ≃)

1

2∫ d𝐸 ]coth

𝐸 − 𝜇

2𝑇+ coth

𝐸 + 𝜇

2𝑇 ≃ 𝑇 ]log sinh

𝐸 − 𝜇

2𝑇+ log sinh

𝐸 + 𝜇

2𝑇

≃ 𝐸 + 𝑇 log(1 + 𝑒−(𝐸−𝜇)⇑𝑇 ) + 𝑇 log(1 + 𝑒−(𝐸+𝜇)⇑𝑇 ) .

(3.21)

Thus the grand potential becomes

Ω

𝑉= ∫

d3k

(2𝜋)3[𝐸k + 𝑇 log(1 − 𝑒−(𝐸k−𝜇)⇑𝑇 ) + 𝑇 log(1 − 𝑒−(𝐸k+𝜇)⇑𝑇 )⌉ , (3.22)

which is apparently the sum of the free energy of the real scalar eld (2.15) for a particle (theenergy is reduced by the chemical potential in the thermal contribution) and for an antiparticle(the energy is increased). The vacuum term is independent of the chemical potential.


3.3 Bose-Einstein condensation

By looking at (3.22) we see that we have a problem if for some k, 𝜇 = 𝐸k is reached. This canhappen if 𝜇 = 𝑚. In that case the thermal contribution blows up. Where does this singularitycome from? By going back to (3.17) we see that at 𝜇 = 𝑚, k = 0 the real part of the coecient inthe exponent becomes zero. Then the integral over 𝜑 in 𝒵GC does not converge. In the languageof (3.18), the operator 𝐷 has a zero mode so that the inverse determinant is innite.

Clearly, we need to separate this zero mode. It turns out that we need to keep the volume niteand the temperature nonzero for the following discussion so that both the Matsubara-frequenciesand the spatial momenta are discrete. We will see that for specic values of the parameters 𝜇and 𝑇 , a nite fraction of the particles resides in this zero-mode state. This is the denition of acondensate.

The density (the negative of the 𝜇-derivative of (3.22)) separated into the zero-mode and ev-erything else reads

𝑛 =1

𝑉]

1

𝑒𝛽(𝑚−𝜇) − 1−

1

𝑒𝛽(𝑚+𝜇) − 1 +

1

𝑉∑k≠0

]1

𝑒𝛽(𝐸k−𝜇) − 1−

1

𝑒𝛽(𝐸k+𝜇) − 1

=1

𝑉]

1

𝑒𝛽(𝑚−𝜇) − 1−

1

𝑒𝛽(𝑚+𝜇) − 1 + ∫

d3k

(2𝜋)3]

1

𝑒𝛽(𝐸k−𝜇) − 1−

1

𝑒𝛽(𝐸k+𝜇) − 1 .

(3.23)

Note that the zero-mode contribution comes with an inverse volume factor. The rest is a sum overall modes except for k = 0. For large volumes the exclusion of one mode makes a negligible dierenceso that we can replace the sum by the integral over k. At 𝜇 = 𝑚 the zero-mode contribution isill-dened so we must treat this as a limit 𝜇→𝑚.

Instead of calculating 𝑛(𝜇,𝑚,𝑇, 𝑉 ) we can simplify the discussion if we go back to the canonicalensemble, x the total density 𝑛 = 𝑛fix and calculate the chemical potential that is necessary toachieve this density, 𝜇(𝑛fix,𝑚,𝑇, 𝑉 ). So let us imagine a situation where we put a xed number ofparticles 𝑛fix ⋅ 𝑉 in our box and gradually reduce the temperature. What happens is the following:

1. For suciently high temperatures, all of the particles reside in the thermal excitations andthere is no condensate (irrespective of how many particles we try to squeeze in our box). Thiscorresponds to a chemical potential 𝜇 <𝑚.

2. As we reduce the temperature, the chemical potential increases. At 𝑇 = 𝑇𝑐(𝑛fix) it hits 𝜇 =𝑚,and here the thermal part of the density saturates and can not accommodate more particles.

3. Reducing the temperature below 𝑇𝑐, 𝑛cond starts to contribute to keep 𝑛fix = 𝑛therm + 𝑛condxed. The critical temperature reads (in an asymptotic expansion, see below)

𝑇𝑐(𝑛) ∼2𝜋

𝑚(

𝑛fix𝜁(3⇑2)

)

2⇑3

. (3.24)

This is visualized in Fig. 3.1. We will now prove each of the above statements.

3.3.1 Thermal density at high temperature

For the rst one, we need to show that the thermal part of the density can be arbitrarily large if 𝑇is suciently high. Throughout we will assume that the mass is positive, 𝑚 = ⋃𝑚⋃. The density isrewritten as → Exercise

𝑛therm =𝑚3

8𝜋2

∞

∑𝑙=1

sinh(𝛽𝜇𝑙)1

2𝜋𝑖∫

𝑐+𝑖∞

𝑐−𝑖∞d𝑧 (

𝛽𝑚𝑙

2)

−𝑧

Γ(𝑧 + 1

2)Γ(

𝑧 − 3

2) , (3.25)

3.3. BOSE-EINSTEIN CONDENSATION 25

Figure 3.1: The thermal part of the density against the chemical potential during the cooling ofthe system. The total density 𝑛fix is kept xed and the chemical potential becomes the function𝜇(𝑛fix, 𝑇 ).

and the high-temperature expansion of this expression reads

𝑛therm𝑇→∞ÐÐÐ→

𝜇𝑇 2

3. (3.26)

Therefore, 𝑛therm increases with the temperature without bounds so that whatever high 𝑛fix is,we can always nd a temperature and chemical potential 𝜇 <𝑚 so that 𝑛therm = 𝑛fix.

3.3.2 Finiteness of the thermal density at the onset

We have seen that the thermal contribution can accommodate an arbitrary density. What happensif we start to lower 𝑇 but want to keep 𝑛therm = 𝑛fix constant? Obviously from (3.26), we need toincrease 𝜇. The maximal allowed value is 𝜇 =𝑚. If 𝑛therm is nite at this point, then this will markthe onset of condensation, where 𝑛cond has to start to contribute. If 𝑛therm is innite, then there isno condensation since we can just put all the density in the thermal part.

So we need to analyze 𝑛therm at 𝜇 → 𝑚. To this end we express (3.25) via a modied Besselfunction, cf. [4] (10.32.13),

1

2𝜋𝑖∫

𝑐+𝑖∞

𝑐−𝑖∞d𝑧 (

𝛽𝑚𝑙

2)

−𝑧

Γ(𝑧 + 1

2)Γ(

𝑧 − 3

2) =

8

𝛽𝑚𝑙⋅𝐾2(𝛽𝑚𝑙) , (3.27)

which gives altogether

𝑛therm =𝑚2

𝜋2𝛽

∞

∑𝑙=1

sinh(𝛽𝜇𝑙)

𝑙𝐾2(𝛽𝑚𝑙) . (3.28)

All terms in the sum are nite. The only way this expression can be innite is if the sum does notconverge. We thus need to take a look at the limit 𝑙 →∞, cf. [4] (10.25.3),

𝐾𝜈(𝛽𝑚𝑙)𝑙→∞ÐÐ→

𝜋

2𝛽𝑚𝑙𝑒−𝛽𝑚𝑙, sinh(𝛽𝜇𝑙)

𝑙→∞ÐÐ→

𝑒𝛽𝜇𝑙

2, (3.29)


where the rst relation holds for any 𝜈. Using the asymptotic behavior of the summand thereforegives

𝑛therm =𝑚2𝑇

𝜋2

𝜋

2𝛽𝑚

∞

∑𝑙=1

𝑙−3⇑2 𝑒𝛽(𝜇−𝑚)𝑙+ terms more convergent . (3.30)

Now we can take the limit 𝜇→𝑚,

𝑛therm(𝜇 =𝑚) = (𝑚𝑇

2𝜋)

3⇑2

𝜁(3⇑2) + nite , (3.31)

where 𝜁(3⇑2) ≈ 2.6 appeared. Since the least convergent part of the expansion is nite, the remainingterms are also nite. Thus 𝑛therm has a nite limit at 𝜇 → 𝑚 at a given temperature. This is themaximal density that the thermal contribution can accommodate.

3.3.3 Critical temperature and order of the transition

Reducing the temperature further is only possible if 𝑛cond > 0, that is to say, if condensation occurs.We already derived an implicit equation for 𝑇𝑐 in (3.31)

𝑛fix ∼ (𝑚𝑇𝑐2𝜋

)

3⇑2

𝜁(3⇑2) , (3.32)

which is based on the asymptotic expansion (3.29). From this it follows

𝑇𝑐(𝑛fix) ∼2𝜋

𝑚(

𝑛fix𝜁(3⇑2)

)

2⇑3

. (3.33)

The density in the condensed mode is obtained by subtracting the thermal contribution from thefull density,

𝑛cond(𝑛fix, 𝑇 ) ∼ ⌊𝑛fix − (𝑚𝑇

2𝜋)

3⇑2

𝜁(3⇑2) ⋅Θ(𝑇𝑐 − 𝑇 ) = 𝑛fix ⋅

⎨⎝⎝⎝⎝⎪

1 − (𝑇

𝑇𝑐(𝑛fix))

3⇑2⎬⎠⎠⎠⎠⎮

⋅Θ(𝑇𝑐 − 𝑇 ) , (3.34)

Figure 3.2: The condensed part of the density against the temperature.

3.3. BOSE-EINSTEIN CONDENSATION 27

which is plotted in Fig. 3.2. The density is continuous at 𝑇𝑐, it is zero in the normal phase andpositive in the condensed phase. The derivative 𝜕𝑛cond⇑𝜕𝑇 is discontinuous at 𝑇𝑐. This is a secondderivative of Ω thus the phase transition is of second order.

3.3.4 Spontaneous symmetry breaking

We have seen that in the condensed phase, the zero-mode carries a nite fraction of the total densityin the innite volume limit. Let us separate the zero mode of the eld as 𝜑 = 𝜑 + 𝜑′. In terms ofthe Fourier components (2.7), these read

𝜑 = 𝜑(0,0), 𝜑′ =𝑇

𝑉∑

(𝑛,k)≠(0,0)

𝜑(𝜔𝑛,k) 𝑒𝑖𝜔𝑛𝜏+𝑖kx , (3.35)

where in the sum the term 𝑛 = k = 0 is excluded, so that ∫ d𝜏 d3x𝜑′ = 0. Then, the action reads

𝑆 = ∫

𝛽

0d𝜏 ∫ d3x )(𝜕𝜏 − 𝜇)𝜑

∗(𝜕𝜏 + 𝜇)𝜑 + 𝜕𝑖𝜑

∗𝜕𝑖𝜑 +𝑚2𝜑∗𝜑⌈

= ∫

𝛽

0d𝜏 ∫ d3x )(𝑚2

− 𝜇2)𝜑∗𝜑 + (𝜕𝜏 − 𝜇)𝜑′∗(𝜕𝜏 + 𝜇)𝜑

′+ 𝜕𝑖𝜑

′∗𝜕𝑖𝜑′+𝑚2𝜑′

∗𝜑′⌈ .

(3.36)

The terms linear in 𝜑′ have cancelled since the space-time-integral of 𝜑′ vanishes. The rst termin (3.36) is the contribution of the zero mode. The rest is what we have calculated in (3.22), withthe exception that 𝜑′ has no zero mode. It gives a path integral 𝒵GC = (det𝐷′)−1. We can imaginethat 𝐷′ has the same eigenvalues as 𝐷 except for the zero mode: 𝑚2 −𝜇2 is replaced by unity as aneigenvalue. For large volumes this change makes a negligible dierence so that we may approximatethe so obtained grand potential with Ω = −𝑇 log𝒵GC.

For a moment we now switch on an innitesimal interaction 𝜆(𝜑∗𝜑)2 and consider its eect forthe zero mode. We have altogether

𝒵GC = ∫

∞

−∞d𝜑 exp ⌊−

Ω𝜑

𝑇−

Ω

𝑇 Ω𝜑 = 𝑉 ((𝑚2

− 𝜇2)𝜑∗𝜑 + 𝜆(𝜑∗𝜑)2) . (3.37)

For 𝜆 > 0 the integral over 𝜑 converges. We will refer to Ω𝜑 as the eective potential for 𝜑. Takingthe negative 𝜇-derivative of the logarithm of Eq. (3.37), we see that the zero-mode contributionseparates from the rest, just as we wrote above in (3.23). The particle number density coming fromthe zero mode is

𝑛cond = 2𝜇 ∫d𝜑𝜑∗𝜑𝑒−Ω𝜑⇑𝑇

∫ d𝜑𝑒−Ω𝜑⇑𝑇

= 2𝜇 ∐𝜑∗𝜑 . (3.38)

The eective potential for 𝜑 has a non-trivial form. For ⋃𝜇⋃ < 𝑚 it has a single minimum at𝜑 = 0, and at that minimum 𝑛cond = 0. However, for ⋃𝜇⋃ > 𝑚, the potential is of the Mexican hattype1 with a circle-shaped minimum at which 𝑛cond > 0.

In the path integral, 𝜑 is integrated over. Still, it is useful to consider the regions of the 𝜑-spacethat dominate 𝒵GC. These are around the minimum of Ω𝜑, where 𝜑 uctuates. In the innite

1Later when we consider interacting scalar fields, we will see that 𝜆 > 0 increases the mass of the particles. Inthis case the chemical potential can even exceed the (bare) mass 𝑚 of the particles. We can also give a physicalinterpretation to why we needed to switch on a small interaction. We need to take into account that the particlesin the condensate repel each other (for example due to their electric charges). If we neglect this, infinitely manyparticles would accumulate in the condensate. An infinitesimal repulsive interaction 𝜆 > 0 solves this (unphysical)problem.


volume limit, these uctuations even die out, since the exponent is proportional to 𝑉 . Thus, the𝑉 →∞ limit has the same eect as the classical limit: only the minimum of the potential survives.In view of the symmetry (3.2) of the action, the dominant eld congurations are symmetric (𝜑 = 0)for 𝜇 < ⋃𝑚⋃, whereas they are not symmetric (𝜑 > 0) for 𝜇 > ⋃𝑚⋃. Since the action is in both casesinvariant under the symmetry transformation, the symmetry breaking is spontaneous.

One may compare this to the Ising model, where the spontaneously broken symmetry is theparity 𝑃 . The congurations relevant for the partition function are disordered spins for hightemperature and aligned spins for low temperature. In the ordered phase 𝑃 is intact, while inthe disordered phase it is spontaneously broken by the magnetization 𝑀 . The equivalent of theseconcepts here are the complex rotation symmetry and 𝑛cond. A big dierence is that the symmetryhere is continuous, whereas for the Ising model it is discrete.

One more aspect of the symmetry breaking in a nite volume is relevant here. For the Isingmodel we needed to introduce a small explicit breaking parameter ℎ to distinguish between the twodegenerate minima (all spins up or all spins down). The magnetization directly at ℎ = 0 vanishesalways due to parity symmetry, and can be nonzero only if ℎ → 0 is taken as a limit. Similarly,∐𝜑 = 0 for complex scalar eld theory even for 𝜇 > ⋃𝑚⋃ the symmetry (3.2) ensures that 𝜑 runsaround the circle and averages to zero. The spontaneous breaking can be understood as having asmall explicit breaking parameter 𝜖𝜑 in the action. This tilts the eective potential Ω𝜑 a bit so thatthe minimum is unique. Finally we consider the limit 𝜖 → 0. Thus, having a nonzero expectationvalue for the eld is to be understood as having lim𝜖→0 ∐𝜑𝜖 ≠ 0.

The ground state for breaks the continuous symmetry (3.2) spontaneously. Then, Goldstone'stheorem ensures that a massless particle appears in the spectrum of the theory. (In the Ising modelthere was no massless particle since there the symmetry (6) was discrete, and the Goldstone theoremonly applies for the continuous case.) The Goldstone boson in our case corresponds to a zero-modeuctuation in the eld 𝜑. This we can see from the propagator of 𝜑, which reads

∐𝜑(x, 𝜏)𝜑(x′, 𝜏 ′) ≡ 𝐺(𝜏 − 𝜏 ′,x − x′) = 𝑇∑𝑛∫

d3k

(2𝜋)3𝑒𝑖𝜔𝑛(𝜏−𝜏 ′)+𝑖k(x−x′)

(𝜔𝑛 − 𝑖𝜇)2 + k2 +𝑚2. (3.39)

Indeed, at 𝜇 = 𝑚 the propagator (namely, the 𝑛 = 0 contribution to it) has a pole at k = 0, i.e. itdescribes a massless mode. We will prove (3.39) later when we consider the interacting scalar eld.

3.4 Proper time representation and renormalization

Having separated the zero-mode contribution, we can continue the analysis of the non-condensedpart of the grand potential. Just as for 𝜇 = 0, this quantity needs additive renormalization. To seethis explicitly, we start from (3.19),

Ω = 𝑇∑𝑛∫

d3k

(2𝜋)3log

(𝜔𝑛 − 𝑖𝜇)2 +𝐸2

k

𝑇 2. (3.40)

We use the same technique as in Sec. 2.2 with the Mellin transform, which allows us to perform themomentum integral and the sum over 𝑛. Notice that everything goes through like in the real case,but this time we need 𝜇 <𝑚 in order to ensure that the real part of the argument of the logarithmis positive. The sum over 𝑛 involves this time the chemical potential as well,

∞

∑𝑛=−∞

𝑒−(𝜔𝑛−𝑖𝜇)2𝑠 =1

2⌋𝜋𝑠𝑇

Θ3 (−𝑖𝜇

2𝑇, 𝑒−1⇑(4𝑠𝑇

2)) . (3.41)


Except for this dierence and an overall factor two, the calculation is the same and results in

Ω

𝑉= −

1

16𝜋2∫

∞

0d𝑠

1

𝑠3𝑒−𝑚

2𝑠 Θ3 (−𝑖𝜇

2𝑇, 𝑒−1⇑(4𝑠𝑇

2)) . (3.42)

An important feature of Ω is that it only depends on the square of the chemical potential→ Exercise.This can also be proven using the expansion of the Θ3 function, see [4] 20.6.4. Irrespective of thechemical potential, the zero-temperature limit of the elliptic theta function is unity, so

Ω(𝑇 = 0, 𝜇 = 0)

𝑉= −

1

16𝜋2∫

∞

0d𝑠

1

𝑠3𝑒−𝑚

2𝑠 , (3.43)

which is the same divergence that we regularized in Sec. 2.3. We also know how to renormalizeit: by subtracting the 𝑇 = 0 limit, as we did in Sec. 2.4. The full renormalized grand potential istherefore

Ωren= Ω(𝑇,𝜇) −Ω(𝑇 = 0, 𝜇 = 0) . (3.44)

Notice that the thermal contribution to the density, 𝑛therm did not need any 𝑇 = 0 subtraction. Infact, it vanishes at 𝑇 = 0 automatically, as visible from (3.30) as long as 𝜇 ≤ 𝑚. In other words,the additive divergences are independent of the chemical potential, which is why they cancel whengoing from Ω to 𝑛. This is also apparent from the divergent part (3.43), which is independent of 𝜇.

Altogether, we have seen that the additive divergences in thermodynamic quantities are inde-pendent of 𝑇 and of 𝜇. We conducted the analysis in the free case, but the conclusion also holdsin interacting theories. From a more general point of view, 𝑇 and 𝜇 are infrared parameters: theyaect the low-energy behavior of the theory. The divergences are, on the other hand, ultravioletcharacteristics.


Let us also determine the thermodynamic potential density across the phase transition. The zero-mode gives negligible contribution to Ω⇑𝑉 in the thermodynamic limit. To see this, note that 𝑛condis nonzero for 𝑉 →∞ in the condensed phase. Thus, from (3.23) we see that

𝜇→𝑚 ∶1

𝑉

1

𝑒𝛽(𝑚−𝜇) − 1is nonzero ⇒ 𝑒𝛽(𝑚−𝜇)

− 1 = 𝒪(1⇑𝑉 ) ⇒ 1 − 𝑒−𝛽(𝑚−𝜇)= 𝒪(1⇑𝑉 ) ,

(3.45)which implies that

Ωcond

𝑉=𝑇

𝑉(log(1⇑𝑉 ) + nite⌋

𝑉→∞ÐÐÐ→ 0 . (3.46)

Therefore, the full thermodynamic potential density is

Ω

𝑉=

)⌉⌉⌋⌉⌉]

Ωtherm(𝜇,𝑇 )𝑉 𝑇 > 𝑇𝑐

Ωtherm(𝜇=𝑚,𝑇 )𝑉 𝑇 ≤ 𝑇𝑐 .

(3.47)

A pronounced signal can be seen in the heat capacity

𝑐𝑉 ≡𝜕𝜖

𝜕𝑇⋀𝑛, (3.48)

where the derivative is to be taken at xed particle number. To obtain 𝑐𝑉 , we need various quantities


Figure 3.3: The pressure and the heat capacity of complex scalar eld theory around phase transitionto the Bose-Einstein condensed phase. The xed density is 𝑛fix⇑𝑚

3 = 1.

from the equation of state:

𝑠 = −1

𝑉

𝜕Ω

𝜕𝑇, 𝑝 = −

Ω

𝑉, 𝜖 = −𝑝 + 𝑇𝑠 + 𝜇𝑛fix , (3.49)

where 𝜇(𝑛fix, 𝑇 ) is an explicit function of the temperature. Fig. 3.3 shows the dependence of thepressure and of 𝑐𝑉 on the temperature around the phase transition. Notice the non-analyticity inboth observables. The gure also includes the Stefan-Boltzmann limits for 𝑝 and for 𝑐𝑉 → Exercise.In Fig. 3.4 we also show the dependence of the thermal part of the pressure on the chemical potential,evaluated numerically.

Figure 3.4: The pressure of charged bosons at nonzero temperatures and chemical potentials.

3.6. CHARGED SCALARS IN LOWER DIMENSIONS 31

3.6 Charged scalars in lower dimensions

It turns out that the condensation phenomenon depends crucially on the space-time dimensionality.In particular, it is instructive to repeat the calculation in 2 + 1 dimensions. The spatial volume ofthe system is denoted by 𝑉2. The thermodynamic potential density is now, comparing to (3.19)

Ω2+1

𝑉2= ∫

d2k

(2𝜋)2𝑇

∞

∑𝑛=−∞


2 +𝐸2k

𝑇 2. (3.50)

We need to do almost the same calculation as in Sec. 3.3.1, just this time the momenta are two-dimensional → Exercise. We get

𝑛2+1therm =𝑚𝑇

2𝜋∑𝑙

𝑙−1 + terms less divergent . (3.51)

Since this involves 𝜁(1) it is innite unless the temperature is exactly zero. Thus, in two spatialdimensions no condensate can form for any 𝑇 > 0. Accordingly, spontaneous symemtry breakingcannot take place, since ∐𝜑 remains at zero. The chemical potential can never reach 𝜇 =𝑚, so thepropagator has no pole at k = 0 that is to say, there is no massless Goldstone boson in the theory.The same conclusion also holds in 1 + 1 dimensions at any temperature.

The absence of the spontaneous breaking of a continuous symmetry in 1+ 1 or 2+ 1 space-timedimensions at 𝑇 > 0 is actually general; it is guaranteed by the Mermin-Wagner theorem. A simpleexplanation of the theorem is the following. If there were a Goldstone boson associated with thebreaking i.e., if the chemical potential could reach 𝜇 =𝑚 the propagator of the eld would looklike (compare (3.39) at 𝜇 =𝑚)

𝐺(𝜏,x) = 𝑇∑𝑛∫

d𝐷k

(2𝜋)𝐷𝑒𝑖𝜔𝑛𝜏+𝑖kx

𝜔2𝑛 + k2 − 2𝑖𝜔2

𝑛𝑚. (3.52)

Let us assume that the propagator is regulated in the ultraviolet (for example by a cuto ⋃k⋃ < Λ)and concentrate on the infrared (small 𝑘) behavior of the integral. For any 𝑛 > 0 the denominatorhas no zero so the integral converges. However, for 𝑛 = 0 the integrand has a pole at k = 0.Separating the 𝑛 = 0 contribution we have

𝐺(𝜏,x) = 𝑇 ∫d𝐷k

(2𝜋)𝐷𝑒𝑖kx

k2+ infrared nite . (3.53)

For 𝐷 ≥ 3 the two-point function is infrared nite, whereas it diverges in 𝐷 ≤ 2. Such a divergenceis unphysical. Our assumption that 𝜇 =𝑚 can be reached or, equivalently, that there is a masslessGoldstone boson must be incorrect. This is the basis of the Mermin-Wagner-Coleman theoremthat forbids symmetry breaking to proceed in one or two spatial dimensions at 𝑇 > 0. If 𝑇 = 0exactly, the sum over the Matsubara frequencies becomes an integral ∫ d𝑘0 so that the divergenceonly occurs if 𝐷 + 1 ≤ 2. Thus, symmetry breaking in 𝐷 = 1 at 𝑇 = 0 is also forbidden.

Chapter 4

Fermion fields

The Lagrangian density for a fermion eld 𝜓 in Minkowski space-time is

ℒ𝑀 = 𝜓(𝑖𝛾𝜇𝜕𝜇 −𝑚1)𝜓 , (4.1)

where 𝜓 = 𝜓𝛾0 and the Dirac matrices satisfy

𝛾𝜇, 𝛾𝜈 = 2𝜂𝜇𝜈 . (4.2)

The momentum conjugate to the eld 𝜓 is

𝜋 =𝜕ℒ𝑀𝜕(𝜕0𝜓)

= 𝜓𝑖𝛾0 = 𝑖𝜓 , (4.3)

implying that 𝜓 and 𝜓 are independent variables. The Hamiltonian is obtained by the Legendretransformation

ℋ = 𝜋𝜕0𝜓 −ℒ𝑀 = 𝜓(−𝑖𝛾𝑘𝜕𝑘 +𝑚1)𝜓 . (4.4)

The spin-statistics theorem tells us that the elds obey anticommutation relations

𝜓𝛼(𝑡,x), 𝜓𝛽(𝑡,y) = 𝜓𝛼(𝑡,x), 𝜓

𝛽(𝑡,y) = 0

𝜓𝛼(𝑡,x), 𝜓𝛽(𝑡,y) = ℎ𝛿(x − y)𝛿𝛼𝛽 .

(4.5)

In the classical limit ℎ → 0, we can replace the eld operators by their eigenvalues 𝑐, 𝑐. Theseare not usual numbers but Grassmann-numbers. Thus, we need to dene the path integral overGrassmannian variables.

4.1 Fermionic oscillator

In this section we follow [2]. For scalar eld theory, we started from the harmonic oscillator, denedusing the ladder operators,

(, ⌋ = 1, (, ⌋ = (, ⌋ = 0 , (4.6)

that appear in the Hamiltonian

=𝜔

2( + ) . (4.7)

32

4.1. FERMIONIC OSCILLATOR 33

The ladder operators act on the eigenstates of as

⋃𝑛 =⌋𝑛 + 1⋃𝑛 + 1, ⋃𝑛 =

⌋𝑛⋃𝑛 − 1 . (4.8)

We also made use of the completeness relations, both for the eigenstates of and of the coordinateand momentum operators,

1 =∑𝑛

⋃𝑛∐𝑛⋃ = ∫ d𝑥⋃𝑥∐𝑥⋃ = ∫d𝑝

2𝜋⋃𝑝∐𝑝⋃ . (4.9)

Now we have to dene these concepts for anticommuting variables, so

, = 1, , = , = 0 . (4.10)

This implies that the states are much simpler here: there is only a vacuum (which is annihilatedby ) and a one-particle state (which is excited from the vacuum by ):

⋃0 ≡ 0, ⋃0 ≡ ⋃1 . (4.11)

To see that there are no other states note that

⋃1 = ⋃0 = (1 − )⋃0 = ⋃0, ⋃1 = ⋃0 = 0 . (4.12)

The analogue of the Hamiltonian is, taking into account the anticommutativity,

=𝜔

2( − ) = 𝜔( − 1⇑2) . (4.13)

4.1.1 Energy representation

Since there are only two states, the energy representation of the partition function is very simple,

𝒵 = tr 𝑒−𝛽 = ∐0⋃𝑒−𝛽 ⋃0 + ∐1⋃𝑒−𝛽 ⋃1 . (4.14)

The eigenvalues of the Hamiltonian on the states are

⋃0 = −𝜔

2, ⋃1 =

𝜔

2, (4.15)

implying that

𝒵 = 𝑒𝛽𝜔⇑2 + 𝑒−𝛽𝜔⇑2 = 2 cosh(𝜔

2𝑇) . (4.16)

Again, the complete equation of state can be derived from this, similarly as in Sec. 1.4. In particular,the free energy is

𝐹 = −𝑇 log𝒵 = −𝑇 log [𝑒𝛽𝜔⇑2(1 + 𝑒−𝛽𝜔)⌉ = −𝜔

2)])𝐹vac

−𝑇 log(1 + 𝑒−𝛽𝜔))⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊]⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊)

𝐹therm

(4.17)

to be compared to the corresponding bosonic free energy, (1.35). Note the overall minus sign thatips the sign of the vacuum contribution. (The thermal contribution is negative like for bosonssince the argument of the logarithm is this time above unity.) The other dierence is the + insidethe logarithm, which is a reminiscent of the Fermi-Dirac statistics.

34 CHAPTER 4. FERMION FIELDS

4.1.2 Path integral

Even though the energy representation is very simple in the present case, it is much more useful tohave the path integral (coordinate state) representation for the same object. To that end we needto dene the completeness relation for Grassmannian variables. Remember for bosons we had ⋃𝑥and ⋃𝑝: the eigenstates of the operators in the Hamiltonian. Thus, we need the eigenstates of theoperators and .

The eigenstates of and of read

⋃𝑐 ≡ 𝑒−𝑐

⋃0, ∐𝑐⋃ ≡ ∐0⋃𝑒−𝑐∗

, (4.18)

and the representation of the trace as well as the completeness relation can be worked out in termsof these states → Exercise.

The path integral can be derived using the same technique as in Sec. 1.2 for bosons and thenal result reads → Exercise,


∫ d𝑐∗𝑁d𝑐𝑁 . . .d𝑐∗1d𝑐1 𝑒

−𝑆 , 𝑆 = 𝜖𝑁

∑𝑖=1

(𝑐∗𝑖+1𝑐𝑖+1 − 𝑐𝑖

𝜖+𝐻(𝑐∗𝑖+1, 𝑐𝑖))𝑐∗𝑁+1=−𝑐

∗1

𝑐𝑁+1=−𝑐1

, (4.19)

revealing that the variables obey antiperiodic boundary conditions in Euclidean time. Again wecan write our result in a continuum form

𝒵 = ∫ 𝑐(𝛽)=−𝑐(0)𝑐∗(𝛽)=−𝑐∗(0)

𝒟𝑐∗(𝜏)𝒟𝑐(𝜏) exp ⌊−∫𝛽

0d𝜏 (𝑐∗(𝜏)

𝜕𝑐(𝜏)

𝜕𝜏+𝐻(𝑐∗(𝜏), 𝑐(𝜏))) . (4.20)

4.2 Path integral

We can generalize (4.20) by replacing the fermionic oscillator Hamiltonian by the spatial integralof (4.4). In essence we need to substitute 𝜓 for 𝑐 and 𝜓 for 𝑐∗ the latter means 𝜓𝛾0 for 𝑐∗. Whatwe obtain in the exponent is thus

𝑆 = ∫

𝛽

0d𝜏 ∫ d3x (𝜓𝛾0𝜕𝜏𝜓 + 𝜓(−𝑖𝛾

𝑘𝜕𝑘 +𝑚1)𝜓) = ∫𝛽

0d𝜏 ∫ d3x𝜓(𝛾0𝜕𝜏 − 𝑖𝛾𝑘𝜕𝑘 +𝑚1)𝜓 . (4.21)

Note that this is the same prescription that we had for the scalar eld theory case: the exponentcontains the Euclidean Lagrangian ℒ that is obtained as ℒ = −ℒ𝑀(𝜏 = 𝑖𝑡). Nevertheless, we cansimplify the Euclidean Lagrangian further if we introduce the Euclidean Dirac matrices

𝛾𝐸0 ≡ 𝛾0, 𝛾𝐸𝑖 ≡ −𝑖𝛾𝑖, 𝛾𝐸𝜇 , 𝛾𝐸𝜈 = 2𝛿𝜇𝜈 , (4.22)

and denote 𝜕0 = 𝜕𝜏 and 𝛾𝜇𝑎𝜇 = ⇑𝑎. We also drop the superscript 𝐸. Thus, the nal expression forthe path integral is

𝒵 = ∫ 𝒟𝜓(𝜏,x)𝒟𝜓(𝜏,x) exp ]−∫𝛽

0d𝜏 ∫ d3xℒ , ℒ = 𝜓( ⇑𝜕 +𝑚1)𝜓 . (4.23)

The integration variables 𝜓 and 𝜓 obey antiperiodic boundary conditions in imaginary time, whichfollows from (4.20).

Following our strategy for the bosonic case, we proceed by Fourier-transforming the elds. Theanalogue of (2.7) is

𝜓(𝜏,x) = 𝑇∑𝑛∫

d3p

(2𝜋)3𝜓(𝜔𝑛,p) 𝑒

𝑖𝜔𝑛𝜏+𝑖px , (4.24)


and taking the complex conjugate, for 𝜓:

𝜓(𝜏,x) = 𝑇∑𝑛∫

d3p

(2𝜋)3𝜓(𝜔𝑛,p) 𝑒

−𝑖𝜔𝑛𝜏−𝑖px . (4.25)

However, now the antiperiodicity implies that

𝜓(𝛽,x) = −𝜓(0,x) ⇒ 𝜔𝑛𝛽 = 2𝜋(𝑛 + 1⇑2) . (4.26)

The 𝜔𝑛 are the fermionic Matsubara frequencies. Note that they are always nonzero. Since thezero Matsubara mode for scalars was the root of all infrared problems, we can anticipate that forfermions no such complications will occur.

Summarizing, the antiperiodic boundary conditions followed from the representation of the tracewith Grassmannian variables, which involved ∐−𝑐⋃ . . . ⋃𝑐. In turn, the use of Grassmannian variableswas motivated by the fact that we want to have elds that satisfy anticommutation relations (4.5).Finally, this is a direct consequence of the Pauli principle.

For the sake of conciseness, we abbreviate

𝜏,x = 𝑥, 𝜔𝑛,p = 𝑝𝜏 ,p = 𝑝, 𝑝𝑥 = 𝑝𝜏𝜏 + px , (4.27)

and also the sum-integrals over Matsubara frequencies and momenta, and integrals over coordinates

⨋𝑝= 𝑇∑

𝑝𝜏

1

𝑉∑p

𝑉→∞ÐÐÐ→ 𝑇∑

𝑝𝜏∫

d3p

(2𝜋)3, ∫

𝑥= ∫

𝛽

0d𝜏 ∫ d3x . (4.28)

In the Fourier components, the action reads

𝑆 = ∫𝑥⨋

𝑝,𝑘𝑒𝑖(𝑝−𝑘)𝑥 𝜓(𝑘)(𝑖⇑𝑝 +𝑚)𝜓(𝑝) = ⨋

𝑝

𝜓(𝑝)(𝑖⇑𝑝 +𝑚)𝜓(𝑝) . (4.29)

Just like for the charged scalar eld, all modes are independent. Changing to the Fourier-transformedvariables in the integration induces a Jacobian that only aects the normalization of 𝒵 and we ig-nore this here. Denoting the matrix in between the Fourier modes in the exponent by 𝑀 = 𝑖⇑𝑝 +𝑚,we can perform the Grassmannian integration. The analogue of (2.18) is

𝒵 = ∫ 𝒟𝜓(𝜏,x)𝒟𝜓(𝜏,x) exp [− 𝜓𝑀𝜓⌉ = 𝐶 ⋅ det(𝑀) , (4.30)

which is basically the generalization of the Grassmannian Gaussian integral → Exercise. Note that𝑀 is a big matrix, it contains all possible Fourier components, indexed by 𝑝, and each of thesecomponents is a 4 × 4 block in Dirac space.


As with scalars, the next step is to derive the proper time representation for 𝑓 = −𝑇 log𝒵⇑𝑉 .However, this time the determinant is not only over momenta and Matsubara frequencies but alsoover Dirac space. First we need to get rid of the Dirac matrices. This we can achieve by introducing

𝛾5 = 𝛾0𝛾1𝛾2𝛾3, 𝛾5, 𝛾𝜇 = 0, 𝛾25 = 1, det𝛾5 = 1 . (4.31)

Using 𝛾5, we can rewrite the determinant as

det𝑀 = det(𝑖⇑𝑝 +𝑚) = det((𝑖⇑𝑝 +𝑚)𝛾5𝛾5) = det(𝛾5(𝑖⇑𝑝 +𝑚)𝛾5) = det(−𝑖⇑𝑝 +𝑚) = det𝑀 , (4.32)


where we exploited the cyclicity of the determinant. This means that the determinant is real (infact it is also positive), so we can write

det𝑀 =⌈

det(𝑀 𝑀) =

⌉

det(⇑𝑝2 +𝑚2) = )𝑝2 +𝑚2⌈2. (4.33)

Here, in the last step we used ⇑𝑝2 = 𝑝21 and took into account the four-fold degeneracy det(𝐴1) = 𝐴4.

Eq. (4.33) can also be checked explicitly → Exercise. Inserting this in the free energy density andrewriting 𝑝2 = 𝜔2

𝑛 + p2, we obtain

𝑓 = −𝑇

𝑉log𝒵 = −2⨋

𝑝log

𝜔2𝑛 +𝐸

2p

𝑇 2, 𝐸2

p = p2+𝑚2 , (4.34)

which is valid up to 𝐸p-independent terms.All in all, we managed to explicitly calculate the determinant over the Dirac matrices, and as a

byproduct, also obtained the determinant as a product of explicitly positive terms. This allows usto use the Mellin transform machinery from Sec. 2.2. There are basically three dierences. First,the overall minus sign that we already encountered in Sec. 4.1.1. Second, the four-fold multiplicitythat induced the prefactor 2 in 𝑓 , which is due to the four components of the Dirac eld 𝜓. Andthird, the fact that now we have the fermionic Matsubara frequencies 𝜔𝑛 = 2𝜋𝑇 (𝑛 + 1⇑2) in thelogarithm. The latter changes the elliptic Θ function part (2.23) of the proper time-integrand:

∞

∑𝑛=−∞

𝑒−(2𝜋𝑇 (𝑛+1⇑2))2𝑠=

1

2⌋𝜋𝑠𝑇

Θ3 (𝜋

2, 𝑒−1⇑(4𝑠𝑇

2)) . (4.35)

This gives rise to a free energy density of

𝑓 =1

8𝜋2∫

∞

0d𝑠

1

𝑠3𝑒−𝑚

2𝑠 Θ3 (𝜋

2, 𝑒−1⇑(4𝑠𝑇

2)) . (4.36)

The appearance of a nonzero rst argument in the elliptic Θ function reminds us of complexscalar elds with a chemical potential. Indeed, comparing to Eq. (3.42), we see that a free chargedboson with chemical potential 𝜇⇑𝑇 = 𝑖𝜋 is exactly equivalent to a free fermion apart from an overallprefactor of −2, which originates from Grassmann integration and the dierence in the number ofdegrees of freedom:

𝑓 fermion(𝑇 ) = −2 ⋅

Ωcharged boson(𝜇 = 𝑖𝜋𝑇, 𝑇 )

𝑉. (4.37)

This has to do with the fact that an imaginary chemical potential can be reinterpreted as a shiftin the boundary condition → Exercise.

4.4 Fermionic and bosonic thermal sums

We just derived the correspondence between charged bosons and fermions. This also relates thehigh-temperature expansions of the two theories to each other. However, in order to derive, forexample, the 𝑇 4 term for fermions from the high-temperature expansion of charged scalars, wewould need all the terms with 𝜇𝑛𝑇 4−𝑛.

The brute-force option is of course to follow the same strategy that we did for the real scalars,namely write the log(1 + 𝑒−𝛽𝐸p) in a series and then use the Mellin and inverse Mellin transforms.However, we can avoid this cumbersome calculation by discovering yet another correspondencebetween fermions and, this time, real scalars. Let us consider a general bosonic and fermionic sum

𝜎𝑏 = 𝑇∑𝑛

𝜙(𝜔𝑏𝑛), 𝜎𝑓 = 𝑇∑

𝑛

𝜙(𝜔𝑓𝑛), 𝜔𝑏

𝑛 = 2𝜋𝑛𝑇, 𝜔𝑓𝑛 = 2𝜋(𝑛 + 1⇑2)𝑇 , (4.38)


where we indicated the bosonic and fermionic Matsubara frequencies by the superscripts on 𝜔𝑛.The free energies are also sums of this form (and also contain dierent prefactors in the two cases,which we will correct for later). The fermionic sum can be rewritten using the bosonic one:

𝜎𝑓(𝑇 ) = +𝑇 (. . . + 𝜙(−3𝜋𝑇 ) + 𝜙(−𝜋𝑇 ) + 𝜙(𝜋𝑇 ) + . . .⌋

= +𝑇 (. . . + 𝜙(−3𝜋𝑇 ) + 𝜙(−2𝜋𝑇 ) + 𝜙(−𝜋𝑇 ) + 𝜙(0) + 𝜙(𝜋𝑇 ) + . . .⌋

− 𝑇 (. . . + 𝜙(−2𝜋𝑇 ) + 𝜙(0) + . . .⌋

= 2𝑇

2]. . . + 𝜙(−6𝜋

𝑇

2) + 𝜙(−4𝜋

𝑇

2) + 𝜙(−2𝜋

𝑇

2) + 𝜙(0) + 𝜙(2𝜋

𝑇

2) + . . .

− 𝑇 (. . . + 𝜙(−2𝜋𝑇 ) + 𝜙(0) + . . .⌋

= 2𝜎𝑏(𝑇 ⇑2) − 𝜎𝑏(𝑇 ) .

(4.39)

Note that the bosonic zero mode cancels in the dierence, in accordance with the fact that fermionshave no zero mode. Note also that one cannot rewrite 𝜎𝑏 in terms of only 𝜎𝑓 : in any combinationof 𝜎𝑓 there is no zero mode, so it cannot be equal to 𝜎𝑏.


For zero-temperature quantities, the above correspondence is very simple. For the vacuum freeenergy densities, we obtain

𝑓vacferm = −4 ⋅ 𝑓vacbos = −4∫d3p

(2𝜋)3𝐸p

2, 𝐸p =

⌈p2 +𝑚2 , (4.40)

which, of course, is only sensible if the same regularization is applied to both sides. For the thermalcontributions, we have

−1

4⋅ 𝑓 thermferm = 2𝑓 thermbos (𝑇 ⇑2) − 𝑓 thermbos (𝑇 ) = ∫

d3k

(2𝜋)3]2𝑇

2log(1 − 𝑒−2𝐸k⇑𝑇 ) − 𝑇 log(1 − 𝑒−𝐸k⇑𝑇 )

= ∫d3k

(2𝜋)3𝑇 log

1 − 𝑒−2𝐸k⇑𝑇

1 − 𝑒−𝐸k⇑𝑇.

(4.41)

Therefore, simplifying the ratio via (1 − 𝑥2)⇑(1 − 𝑥) = (1 + 𝑥), we obtain

𝑓 thermferm = −4𝑇 ∫d3p

(2𝜋)3log(1 + 𝑒−𝐸p⇑𝑇 ) , (4.42)

and for the full (vacuum plus thermal) free energy

𝑓ferm = −4∫d3p

(2𝜋)3]𝐸p

2+ 𝑇 log(1 + 𝑒−𝐸p⇑𝑇 ) , (4.43)

which conrms what we obtained for the fermionic oscillator (4.17).Finally, the high-temperature expansion from (2.40) reads

−1

4𝑓 thermferm = −

𝜋2

45

𝑇 4

16+

1

12𝑚2𝑇

2

4−

1

6𝜋⋃𝑚⋃

3𝑇

2−𝑚4

32𝜋(2𝛾𝐸 −

3

2+ 2 log

𝑚

2𝜋𝑇)

+𝜋2

90𝑇 4

−1

24𝑚2𝑇 2

+1

12𝜋⋃𝑚⋃

3𝑇 +𝑚4

64𝜋2(2𝛾𝐸 −

3

2+ 2 log

𝑚

4𝜋𝑇) ,

(4.44)


which simplies to

𝑓 thermferm = −7𝜋2

180𝑇 4

+1

12𝑚2𝑇 2

+𝑚4

16𝜋2(2𝛾𝐸 −

3

2+ 2 log

𝑚

𝜋𝑇) +𝒪(𝑇 −2) . (4.45)

Note the disappearance of the odd term ⋃𝑚⋃3𝑇 which as we learned in the meantime originatesfrom the zero mode of scalar elds. It is absent here since fermions have no zero modes, thus the𝑓ferm is an analytic (and even) function of the mass.

Figure 4.1: High-temperature expansion of the pressure of free fermions, compared to the full result.

4.6 Nonzero chemical potential

As for the complex scalar eld in Chap. 3, we can also introduce a nonzero chemical potential forfermions. The Minkowskian Lagrangian is our starting point:

ℒ𝑀 = 𝜓(𝑖 ⇑𝜕 −𝑚)𝜓 . (4.46)

Recall that we need a continuous U(1) symmetry, which we can identify:

𝜓 → 𝑒𝑖𝛼𝜓, 𝜓 → 𝜓 𝑒−𝑖𝛼 . (4.47)

The corresponding Noether current reads

𝑗𝜇 =𝜕ℒ𝑀𝜕(𝜕𝜇𝜓)

𝜕𝜓

𝜕𝛼⋀𝛼=0

= −𝜓𝑖𝛾𝜇 ⋅ 𝑖𝜓 = 𝜓𝛾𝜇𝜓 . (4.48)

Note that the derivative with respect to 𝜕𝜇𝜓 anticommutes with 𝜓 to produce the minus sign above.The fermion number is

𝑁 = ∫ d3x 𝑗0 = 𝜓𝛾0𝜓 . (4.49)

We proceed like in Sec. 3.2 by adding −𝜇 as an operator to the Hamiltonian. This time thenumber operator 𝑁 does not contain a momentum operator 𝜕0𝜓, so the derivation of the path

4.6. NONZERO CHEMICAL POTENTIAL 39

integral remains as simple as at 𝜇 = 0. We just need to shift 𝑉 (𝜓,𝜓) by −𝜇𝑁 . This amounts tothe replacement 𝜕𝜏 → 𝜕𝜏 − 𝜇 in (4.21). All in all we observe again that the simple derivative needsto be promoted to the covariant derivative 𝐷𝜈 = 𝜕𝜈 − 𝜇𝛿𝜈,0. In the Fourier-transformed variant thiscauses the shift 𝑖𝜔𝑛 → 𝑖𝜔𝑛 − 𝜇, or, 𝜔𝑛 → 𝜔𝑛 + 𝑖𝜇. This is the same shift that we had for chargedbosons up to a sign of 𝜇, which is irrelevant since 𝒵GC was a function of 𝜇2 only as we mentionedafter (3.42).

All in all, the thermodynamic potential can again be calculated using the trick (4.39) withΩ for charged bosons. The overall prefactor is −2. So using the trick on (3.22), with the samemanipulations as in (4.41), we arrive at

Ωthermferm

𝑉= −2𝑇 ∫

d3p

(2𝜋)3[log(1 + 𝑒−(𝐸p−𝜇)⇑𝑇 ) + log(1 + 𝑒−(𝐸p+𝜇)⇑𝑇 )⌉ , (4.50)

for the thermal part. The vacuum part is independent of 𝜇.

Chapter 5

Background electromagnetic fields

Besides the temperature and the chemical potential, another parameter that plays a relevant rolein high-energy particle physics, is a background electromagnetic eld. It is therefore of interest toderive the dependence of the free energy on the background eld.

We have seen that due to virtal particle uctuations, the free energy receives contributionsalready in the vacuum: at zero temperature and zero density. However, these contributions are allunphysical (divergent) and cancel in observables. When electromagnetic elds couple to chargedparticles, this is not true anymore: we will see that observables are eld-dependent even in thevacuum. A simple argument that enlightens why this needs to be so is via considering a virtualparticle-antiparticle pair that pops out of the vacuum for some short time, see Fig. 5.1.

Figure 5.1: A virtual particle-antiparticle pair couples to a magnetic (left) or to an electric (right)background eld in the center-of-mass frame.

For a magnetic eld B, the particles with some initial momentum are forced on circular orbitsand thus generate angular momentum parallel to the magnetic eld. This carries an observableeect. For electric elds E the eect is even more drastic: the virtual particles are pulled apartto potentially macroscopic distances so that they can become real particles. This is the so-calledSchwinger eect. In this chapter we explore these eects in more detail.

5.1 Path integral

Here we do not derive the path integral representation for QED but only show the nal result. Thedetails of the derivation can be found in [5] or in [2]. Just as for scalars or for fermions, 𝒵 containsthe Euclidean Lagrangian,

𝒵 = ∫ 𝒟𝐴𝜇𝒟𝜓𝒟𝜓 exp ]−∫𝛽

0d𝜏 ∫ d3xℒ , ℒ =

𝐹 2𝜇𝜈

4+ 𝜓( ⇑𝐷 +𝑚)𝜓 . (5.1)

40

5.2. MINKOWSKIAN AND EUCLIDEAN SPACE-TIMES 41

A subtlety in the derivation concerns the timelike component of the gauge eld: ℒ contains 𝐴𝜏 =

−𝑖𝐴0.In (5.1), 𝐷𝜇 is the covariant derivative

𝐷𝜇 = 𝜕𝜇 − 𝑖𝑞𝐴𝜇 , (5.2)

containing the electric charge 𝑞 of the fermion. Through 𝐷𝜇, the eld strength tensor can beexpressed as

𝐹𝜇𝜈 =𝑖

𝑞(𝐷𝜇,𝐷𝜈⌋ . (5.3)

If the electromagnetic eld is considered a background eld that does not uctuate, the pathintegral over 𝐴𝜇 becomes trivial. In addition, we assume that the eld is constant in space andtime, so that the action also simplies,

𝒵(𝐹𝜇𝜈) = ∫ 𝒟𝜓𝒟𝜓 exp ⌊−𝑉

𝑇

𝐹 2𝜇𝜈

4− ∫

𝛽

0d𝜏 ∫ d3x𝜓( ⇑𝐷 +𝑚)𝜓 . (5.4)

5.2 Minkowskian and Euclidean space-times

The electric and magnetic elds E and B appear in the eld strength tensor as

𝐹𝑖0 = −𝐹0𝑖 = 𝐸𝑖, 𝐹𝑖𝑗 = −𝜖𝑖𝑗𝑘𝐵𝑘 , (5.5)

that is related to the vector potential as

𝐹𝜇𝜈 = 𝜕𝜇𝐴𝜈 − 𝜕𝜈𝐴𝜇 . (5.6)

Going from Minkowskian to Euclidean metric implies 𝑡 = −𝑖𝜏 so that 𝜕𝑡 = 𝑖𝜕𝜏 . In addition, abovewe mentioned that in the Euclidean metric the timelike component of the vector potential needsthe replacement 𝐴0 = 𝑖𝐴𝜏 . This means that E𝑀 = 𝑖E, while B𝑀 = B. Thus, Euclidean electricelds correspond to imaginary Minkowskian electric elds. Magnetic elds, on the other hand, arethe same for both metrics.

In the following we use the Euclidean metric as usual. At (strictly) zero temperature nothingdistinguishes between space and time thus electric and magnetic elds are also equivalent. There-fore it suces to only consider, say, a magnetic eld to derive the thermodynamic observables. Thedrastic dierence between electric and magnetic elds depicted in Fig. 5.1 is then caused onlyby the dierent analytical continuation as we go back to Minkowskian metric.

5.3 Fermions in a magnetic field

Without loss of generality, we can orient the magnetic eld in the positive 𝑧 direction and assumethat the electric charge of the particle 𝑞 is positive. Our choice for the vector potential that generatesthis eld is the so-called Landau gauge,

𝐴𝑦 = 𝐵𝑥, 𝐴𝑥 = 𝐴𝑧 = 𝐴𝜏 = 0 . (5.7)

Unlike in the previous chapters, here we start the discussion with fermions, since as will becomeclear, the bosonic case can be simply obtained from fermions (but not the other way around). TheDirac operator with minimal coupling to the background (5.7) reads

⇑𝐷 = 𝛾𝜇𝐷𝜇 = 𝛾𝜇(𝜕𝜇 − 𝑖𝑞𝐵𝑥𝛿𝜇,𝑦) , (5.8)

42 CHAPTER 5. BACKGROUND ELECTROMAGNETIC FIELDS

which is antihermitean⇑𝐷= − ⇑𝐷 . (5.9)

Notice that, since ⇑𝐷 depends explicitly on 𝑥, the action is not diagonal in Fourier space as it wasfor 𝐵 = 0 in (4.29). Nevertheless, the trick (4.32) with 𝛾5 can still be applied, so that we can derive𝒵 by calculating the square root of the determinant of

𝑀 𝑀 = ( ⇑𝐷 +𝑚)( ⇑𝐷 +𝑚) = − ⇑𝐷

2+𝑚2 , (5.10)

where we used (5.9). Thus, we need the eigenvalues of − ⇑𝐷2.

We begin by rewriting the operator as

⇑𝐷2= 𝛾𝜇𝛾𝜈𝐷𝜇𝐷𝜈 =

1

2((𝛾𝜇, 𝛾𝜈⌋ + 𝛾𝜇, 𝛾𝜈) ⋅

1

2((𝐷𝜇,𝐷𝜈⌋ + 𝐷𝜇,𝐷𝜈) . (5.11)

Using (4.22) and dening the spin operator

𝜎𝜇𝜈 =1

2𝑖(𝛾𝜇, 𝛾𝜈⌋ , (5.12)

and taking into account the properties of the covariant derivative (5.2) we obtain

⇑𝐷2= (𝑖𝜎𝜇𝜈 + 𝛿𝜇𝜈1)

1

2(−𝑖𝑞𝐹𝜇𝜈 + 𝐷𝜇,𝐷𝜈) =𝐷

2𝜇 1 + 𝜎𝑥𝑦𝑞𝐵 , (5.13)

where in the second step we used symmetry-antisymmetry and inserted the magnetic eld 𝐵 =

𝐹𝑥𝑦 = −𝐹𝑦𝑥 from (5.5). We will later need the explicit form of 𝜎𝑥𝑦,

𝜎𝑥𝑦 = (𝜎𝑧 00 𝜎𝑧

) , 𝜎𝑧 = (1 00 −1

) . (5.14)

We write the eigenvalue equation as

⇑𝐷2𝜓 = −𝜆2𝜓 , (5.15)

since ⇑𝐷 is antihermitean, so that 𝜆 ∈ R. The eigenvalues can be obtained analytically and read →Exercise,

𝜆2ℓ,± = 𝑞𝐵(2ℓ + 1 − 2𝑠𝑧) + 𝑝2𝑧 + 𝜔

2𝑛 , (5.16)

We can assign physical meaning to the quantum numbers. The 𝑠𝑧 = ±1⇑2 is the spin of the particleparallel to the magnetic eld, as it also equals the eigenvalue of 𝜎𝑥𝑦, the relativistic spin operator.If the spin is positive, the eigenvalue (energy) is lower, so the magnetic eld tends to align the spinwith itself. The remaining part of the eigenvalue must come from orbital motion: indeed, 2ℓ + 1 isthe quantized angular momentum. Typically, the energy levels characterized by angular momentumundergo a splitting due to spin-contributions. While in usual cases this Zeeman splitting is a smallcorrection, in our case it is just as big as the distance between two consecutive angular momentumlevels, see Fig. 5.2. The levels are called Landau levels. Notice that the lowest Landau level (andonly that) has eigenvalue zero.

Finally, 𝑝2𝑧 and 𝜔2𝑛 in (5.16) comes from the standing waves in the 𝑧 and 𝜏 directions, respectively

(taking into account the antiperiodic boundary conditions in the imaginary time direction). In anite volume 𝑉 = 𝐿𝑥𝐿𝑦𝐿𝑧with periodic boundary conditions, the degeneracy of the eigenvalues isΦ⇑(2𝜋)-fold, where Φ = 𝑞𝐵𝐿𝑥𝐿𝑦 is the ux of the magnetic eld. There is a further factor of 2 dueto particles and antiparticles → Exercise.


Figure 5.2: Visualization of the Landau-levels.


We can write down the analogue of the 𝐵 = 0 formula, (4.34). Exploiting (5.10), the free energydensity is

𝑓 = −𝑇

𝑉log𝒵 = −

𝑇

2𝑉log det(𝑀 𝑀) = −

𝑇

2𝑉∑𝜆

log𝜆2 +𝑚2

𝑇 2, (5.17)

where ∑𝜆 denotes the sum over all eigenvalues and we neglected some energy-independent constantsas usual. Inserting the eigenvalues and their degeneracies, we get

𝑓 = −𝑇

2𝑉⋅ 2 ⋅

Φ

2𝜋∑𝑠𝑧

∑ℓ

∑𝑝𝑧 ,𝑛

log𝑞𝐵(2ℓ + 1 − 2𝑠𝑧) + 𝑝

2𝑧 + 𝜔

2𝑛 +𝑚

2

𝑇 2

𝑉→∞ÐÐÐ→ −

𝑞𝐵𝑇

2𝜋∑

𝑛,ℓ,𝑠𝑧∫

d𝑝𝑧2𝜋

log𝑞𝐵(2ℓ + 1 − 2𝑠𝑧) + 𝑝

2𝑧 + 𝜔

2𝑛 +𝑚

2

𝑇 2

(5.18)

where in the second step we sent the volume 𝑉 = 𝐿𝑥𝐿𝑦𝐿𝑧 to innity, with the momentum-sumconverted into an integral via (4.28) and inserted the ux Φ = 𝑞𝐵𝐿𝑥𝐿𝑦.

Now, besides the formulas we already used for scalars in Sec. 2.2 and for fermions in Sec. 4.3,we also need in addition how to sum over ℓ and over 𝑠𝑧:

∞

∑ℓ=0

𝑒−𝑞𝐵(2ℓ+1)𝑠 =1

2 sinh(𝑞𝐵𝑠), ∑

𝑠𝑧=±1⇑2

𝑒2𝑞𝐵𝑠𝑧 𝑠 = 2 cosh(𝑞𝐵𝑠) . (5.19)

Inserting this into (5.18), we obtain

𝑓 =1

8𝜋2∫

∞

0d𝑠

1

𝑠2𝑒−𝑚

2𝑠 𝑞𝐵 coth(𝑞𝐵𝑠)Θ3 (𝜋

2, 𝑒−1⇑(4𝑠𝑇

2)) . (5.20)

The dierence to the 𝐵 = 0 free energy is the appearance of coth(𝑞𝐵𝑠) due to the sums over ℓ and𝑠𝑧 and the overall factor 𝑞𝐵 due to the degeneracy of the Landau levels. We have already analyzedthe 𝐵 = 0 part in quite some detail, so we simplify the discussion by subtracting that contribution

∆𝑓 ≡ 𝑓(𝐵) − 𝑓(𝐵 = 0) =1

8𝜋2∫

∞

0d𝑠

1

𝑠2𝑒−𝑚

2𝑠]𝑞𝐵 coth(𝑞𝐵𝑠) −

1

𝑠 Θ3 (

𝜋

2, 𝑒−1⇑(4𝑠𝑇

2)) , (5.21)


where we have expanded around 𝐵 = 0 to get coth 𝑞𝐵𝑠 = 1⇑𝑞𝐵𝑠 + 𝑞𝐵𝑠⇑3 +𝒪(𝐵3).

5.5 Renormalization

Since the magnetic eld and the temperature are similar external parameters, it is expected thatultraviolet divergences cancel in the dierence 𝑓(𝐵) − 𝑓(0), just as they do in 𝑓(𝑇 ) − 𝑓(0). Let uscheck this at zero temperature. As we know from (2.26), the Θ3 function equals 1 for 𝑇 = 0. Themagnetic eld-dependence is analytic in the free energy; it can be Taylor-expanded around 𝐵 = 0.This expansion is even in 𝐵, which reects parity symmetry. The 𝒪(𝐵0) term we have alreadysubtracted in ∆𝑓 above. The subsequent terms in the expansion are

∆𝑓(𝐵,𝑇 = 0) =(𝑞𝐵)2

24𝜋2∫

∞

0d𝑠

1

𝑠𝑒−𝑚

2𝑠−(𝑞𝐵)4

360𝜋2∫

∞

0d𝑠 𝑠 𝑒−𝑚

2𝑠+𝒪(𝐵6

) . (5.22)

The 𝒪(𝐵2) term is ultraviolet divergent. Using cuto regularization it is obtained to be

𝑓(𝒪(𝐵2),0) =

(𝑞𝐵)2

24𝜋2⌊log

Λ2

𝑚2− 𝛾𝐸 . (5.23)

Higher orders in 𝐵 all come with a higher power of 𝑠 and are therefore all nite. Having a 𝐵-dependent divergence in the free energy is unphysical changing the physical (IR) parameter 𝐵cannot change the UV behavior of the theory.

We know that in interacting quantum eld theories the parameters of the Lagrangian undergomultiplicative renormalization. They are not the quantities that can be measured in experimentsbut only the bare parameters of the action. For example for scalar eld theory with a 𝜆𝜑4 term, wehave 𝑚2

𝑏 = 𝑍𝑚𝑚2𝑟 , where both 𝑚𝑏 and 𝑍𝑚 are divergent but 𝑚𝑟 is nite. The wave function can also

have a nontrivial renormalization factor, 𝜑𝑏 = 𝑍𝜑𝜑𝑟. While the free energy may be divergent whenexpressed as a function of the bare parameters, it is nite when expressed using the renormalizedparameters.

In the present case we also have an interaction (between the fermion and the background eld).Correspondingly, the coupling renormalizes as 𝑞𝑏 = 𝑍𝑞𝑞𝑟. In addition, the magnetic eld itself alsorenormalizes as 𝐵𝑏 =

⌋𝑍3𝐵𝑟. However, the free energy only depends on the product 𝑞𝐵, and this

combination turns out to be renormalization group invariant due to the QED Ward identity. Inother words, the Ward identity states 𝑍𝑞

⌋𝑍3 = 1 → Exercise. Therefore, we have no freedom to

redene 𝑞𝐵 in the action in order to get rid of the divergence in ∆𝑓 . What did we miss?

Looking back at (5.4) we see that in fact we forgot to consider the contribution to the freeenergy that is carried by the magnetic eld itself,

𝑓𝛾 =𝑇

𝑉∫ d4𝑥

1

4𝐹𝜇𝜈,𝑏𝐹𝜇𝜈,𝑏 =

𝐵2𝑏

2, (5.24)

which is just the classical energy of the magnetic eld. Notice that this energy is independentof the charge of the fermion; it only contains the bare magnetic eld 𝐵𝑏. This means that therenormalization constant of the electromagnetic vector potential

⌋𝑍3 enters here. In addition, this

term is also quadratic in the magnetic eld and thus can be combined with the quadratic part ofthe fermionic free energy. This sum reads

𝑓𝛾 + 𝑓(𝒪(𝐵2),0) =

𝐵𝑏2

2⌊1 + 𝑞2𝑏

1

12𝜋2log

Λ2

𝑚2 . (5.25)

5.6. MAGNETIC SUSCEPTIBILITY 45

We are now free to dene this as the renormalized magnetic eld (squared):

𝐵2𝑏𝑍

−13 = 𝐵2

𝑟 , 𝑍−13 = 𝑍2

𝑞 = 1 + 𝛽1𝑞2 log

Λ2

𝑚2, 𝛽1 =

1

12𝜋2. (5.26)

Then the total (magnetic+matter) free energy to order 𝐵2 equals just 𝐵2𝑟 ⇑2. 𝐵𝑟 is the physical

magnetic eld. It is nite and so is the total free energy. This requirement xes the renormalizationconstant 𝑍3 (and through the Ward identity derived above, also 𝑍𝑞 is xed). The renormalizationconstant agrees with the result of the ususal eld theoretical calculation involving the two-pointfunction for the photon eld (the vacuum polarization diagram), compare, for example (7-18) of [6].The coecient 𝛽1 is the lowest-order 𝛽-function coecient1, that determines whether the theoryis asymptotically free or not. We obtained 𝛽1 > 0, which means (correctly) that QED has noasymptotic freedom.

Altogether, the total free energy at 𝑇 = 0 reads

𝑓(𝐵,0) + 𝑓𝛾 = 𝑓(0,0))⌊⌊⌊⌊⌊⌊⌊]⌊⌊⌊⌊⌊⌊⌊)

irrelevant shift

+ 𝐵2𝑟 ⇑2

)])pure magnetic energy

+ 𝒪((𝑞𝐵)4))⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊]⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊⌊)

matter contribution

. (5.27)

In the following we won't be interested in the pure magnetic energy 𝐵2𝑟 ⇑2, since it is just an

additive constant in the free energy. This means that we have to subtract the term 𝒪(𝐵2) at 𝑇 = 0(i.e. (5.23)) from the free energy. Going to nite temperatures does not change this renormalizationprescription. Altogether our denition of the renormalized free energy reads

∆𝑓𝑟(𝐵,𝑇 ) = 𝑓(𝐵,𝑇 ) − 𝑓(0,0) − 𝑓(𝒪(𝐵2),0). (5.28)

Note that a dependence on the renormalization scheme appears: we could have dened 𝐵𝑟 so thatat 𝑇 = 0 the total free energy is not 𝐵2

𝑟 ⇑2 but 𝐵2𝑟 ⇑2 + const. This changes 𝑍𝑞 by a nite amount

and shifts the free energy by a constant that cancels in dierences of free energies. However, ourscheme is the natural choice: it reproduces the classical energy 𝐵2

𝑟 ⇑2 of the magnetic eld at zerotemperature and low magnetic elds.

Inserting (5.21) and (5.22) into (5.28),

∆𝑓𝑟 =1

8𝜋2∫

∞

0

d𝑠

𝑠3𝑒−𝑚

2𝑠⌊𝑞𝐵𝑠 coth(𝑞𝐵𝑠)Θ3 (

𝜋

2, 𝑒−1⇑(4𝑠𝑇

2)) − 1 −

(𝑞𝐵𝑠)2

3 . (5.29)

5.6 Magnetic susceptibility

Next we will concentrate on low values of the magnetic eld. The response of the system to smallvalues of 𝐵 is encoded in the magnetic susceptibility,

𝜒 = −𝜕2∆𝑓𝑟𝜕(𝑞𝐵)2

⋁𝐵=0

. (5.30)

Note that the rst derivative (magnetization) vanishes at 𝐵 = 0, since there is no preferred directionin the system (∆𝑓𝑟 is an even function of 𝐵). If 𝜒 > 0, the free energy is reduced by the magneticeld, so it is energetically favorable for the system to be exposed to 𝐵. This is a paramagnetic

1Only the lowest order coefficient appears, because the magnetic field has no dynamics, i.e. there is no internalphoton. Thus higher-loop diagrams are absent.


response. The opposite response, 𝜒 < 0 describes diamagnetism: it is energetically favorable to haveno magnetic eld. Both types of materials exist in nature.

Let us calculate whether a gas of fermions at nonzero temperature is para- or diamagnetic. Weneed the second derivative of (5.29) at 𝐵 = 0. This reads

𝜒 = −1

12𝜋2∫

∞

0

d𝑠

𝑠𝑒−𝑚

2𝑠]Θ3 (

𝜋

2, 𝑒−1⇑(4𝑠𝑇

2)) − 1 . (5.31)

The 𝑠-integral is well-dened (the square bracket is exponentially small at low 𝑠) and is calculatednumerically. The result is plotted in Fig. 5.3. Since Θ3(𝜋⇑2, . . .) − 1 is negative, the nal result ispositive for all temperatures thus fermions are paramagnetic.

Below we will learn how to transform the fermionic free energy to the free energy of chargedscalars in the presence of magnetic elds. Anticipating that result, the susceptibility for scalarsreads

𝜒scalar = −1

48𝜋2∫

∞

0

d𝑠

𝑠𝑒−𝑚

2𝑠[Θ3 (0, 𝑒−1⇑(4𝑠𝑇

2)) − 1⌉ . (5.32)

Due to the dierent argument in the elliptic theta function, this time we have Θ3(0, . . .)− 1 > 0, sothat the integral is negative everywhere. Thus, scalar particles are diamagnetic.

Figure 5.3: Magnetic susceptibility of free fermions and free scalars as a function of the temperature.

To have a physical interpretation for this, we can go back to the Landau levels (5.16). Fermionscouple to 𝐵 both via angular momentum and via spin. Scalar particles have no spin; they onlycouple to 𝐵 via angular momentum. Why the latter gives rise to diamagnetism can be understoodin classical terms. Switching on a magnetic eld induces currents that act to reduce the magneticeld (Lenz's law). Thus, the magnetic moment generated by the currents is opposite the externaleld this is diamagnetism.2 Fermions also have spin that tends to align with the magnetic eld,producing a magnetic moment parallel to 𝐵 and, thus, giving paramagnetism. The two phenomenafor fermions can be separated to nd that the spin eect is dominant, so altogether fermions areparamagnetic.

2This argument is too simplistic; in fact we need quantum particles to describe magnetism, cf. the Bohr-vanLeeuwen theorem.

5.7. SCHWINGER EFFECT 47

5.7 Schwinger effect

As described around Fig. 5.1, electric elds can produce real particle-antiparticle pairs, or, in otherwords, they can cause the vacuum to decay. Decay rates have to do with imaginary energies.Remember that for a particle at rest the time evolution of the wave function is given by 𝜓(𝑡) ∝exp(−𝑖𝑚𝑡). If we want do describe a particle that decays, i.e. that the probability ⋃𝜓(𝑡)⋃2 diminishesin time, the mass needs to have an imaginary part. More specically,

⋃𝜓(𝑡)⋃2 ∝ 𝑒−Γ𝑡 → Γ = −2 Im𝑚. (5.33)

In quantum eld theory, the corresponding relation turns out to be very similar [7]. The rate ofdecay of the vacuum to any other state (thus, the probability that particles are created) is

Γ = −2 Im𝐹 . (5.34)

Thus we need to calculate the imaginary part of the free energy 𝐹 = 𝑓 ⋅ 𝑉 .

We start from the renormalized free energy density in a constant (Minkowskian) electric eld 𝐸.The calculation is done at zero temperature; it is simple to generalize it to 𝑇 > 0 as well. The freeenergy is obtained from the corresponding magnetic free energy (5.29) with the replacement 𝐵 →−𝑖𝐸M where 𝐸M denotes the Minkowskian electric eld. In the following we omit the superscriptM. The analytic continuation implies 𝐵 coth𝐵 → 𝐸 cot𝐸, so

𝑓 =1

8𝜋2∫

∞

0

d𝑠

𝑠3𝑒−𝑚

2𝑠⌊𝑞𝐸𝑠 cot(𝑞𝐸𝑠) − 1 +

(𝑞𝐸𝑠)2

3 . (5.35)

For the 𝐵 > 0 case this integrand was well-dened along the real 𝑠-axis. On the contrary, now wehave singularities at

𝑠 =𝑛𝜋

𝑞𝐸, 𝑛 = 1,2, . . . . (5.36)

(At 𝑛 = 0 the square bracket in (5.35) is 𝒪(𝑠4) so the integrand vanishes there.) The cot functionhas simple poles at these points,

𝑞𝐸𝑠 cot(𝑞𝐸𝑠)𝑠→𝑛𝜋

𝑞𝐸ÐÐÐ→

𝑠

𝑠 − 𝑛𝜋𝑞𝐸

+𝒪 (𝑠 −𝑛𝜋

𝑞𝐸) , (5.37)

which can be seen by looking at the limit,

lim𝑠→𝑛𝜋

𝑞𝐸

(𝑠 − 𝑛𝜋𝑞𝐸 ) cos(𝑞𝐸𝑠)

sin(𝑞𝐸𝑠)= lim

𝑠→𝑛𝜋𝑞𝐸

cos(𝑞𝐸𝑠) + (𝑞𝐸𝑠 − 𝑛𝜋) sin(𝑞𝐸𝑠)

𝑞𝐸 cos(𝑞𝐸𝑠)=

1

𝑞𝐸, (5.38)

where we employed the L'Hospital rule.

To make the integral well-dened, we need to shift the poles away from the real axis by adding+𝑖𝜖 to the denominators. This shifting of the poles by +𝑖𝜖 coincides with the Feynman prescriptionfor a Minkowski-spacetime propagator and follows from causality → Exercise. (Up to now we didnot need this because we did not encounter any poles in Euclidean space-time.) Thus we have

𝑓 =1

8𝜋2∫

∞

0

d𝑠

𝑠3𝑒−𝑚

2𝑠⎨⎝⎝⎝⎝⎪

[nite, real] +∞

∑𝑛=1

𝑠

𝑠 − 𝑛𝜋𝑞𝐸 + 𝑖𝜖

⎬⎠⎠⎠⎠⎮

. (5.39)


Since we only need the imaginary part of the free energy, we can concentrate on the sum. Usingthe Cauchy principal value theorem → Exercise

lim𝜖→0+

1

𝑥 + 𝑖𝜖= 𝒫

1

𝑥− 𝑖𝜋𝛿(𝑥) , (5.40)

to take the limit 𝜖→ 0,

Im𝑓 =1

8𝜋2∫

∞

0

d𝑠

𝑠2𝑒−𝑚

2𝑠∞

∑𝑛=1

Im

⎨⎝⎝⎝⎝⎪

𝒫1

𝑠 − 𝑛𝜋𝑞𝐸

− 𝑖𝜋𝛿 (𝑠 −𝑛𝜋

𝑞𝐸)

⎬⎠⎠⎠⎠⎮

= −1

8𝜋

∞

∑𝑛=1

1

𝑠2𝑒−𝑚

2𝑠⋀𝑠=𝑛𝜋

𝑞𝐸

, (5.41)

where we threw away the principal value which, being real, does not contribute to Im𝑓 .

Altogether, the decay rate (5.34) reads

Γ = 𝑉(𝑞𝐸)2

4𝜋3

∞

∑𝑛=1

1

𝑛2exp ⌊−

𝑚2𝑛𝜋

𝑞𝐸 =

(𝑞𝐸)2

4𝜋3Li2 (𝑒

−𝑚2𝜋⇑(𝑞𝐸)) , (5.42)

where Li2 is the dilogarithm function. This coincides with Schwinger's result [7] (𝑓 compares with−ℒ in the notation of [7]). We observe that Γ is exponentially suppressed as long as 𝐸 < 𝐸𝑐 ≡𝑚

2𝜋⇑𝑞,see Fig. 5.4. Thus, roughly speaking, the quantum vacuum becomes unstable above this criticalelectric eld. The situation changes only slightly if we consider pair creation for charged scalarelds (see the gure) we will calculate Γ for this case below. Note that Γ is extensive, since thenumber of produced pairs is proportional to the volume where the electric eld is present.

Figure 5.4: Pair production rate as a function of the electric eld for fermions (red) and for scalars(blue).

Clearly, we could not have found this result if we considered the electric eld as a small per-turbation. Indeed, the Taylor series of Γ(𝐸) vanishes at 𝐸 = 0, or, in other words, Γ(𝐸) is non-perturbative. Physically, the ground state with many macropscopically separated electric chargesis fundamentally dierent from the èmpty' vacuum, so that a small perturbation of the latter (byan innitesimal electric eld) cannot capture the Schwinger eect. Note that if the electric eldis maintained for a long period of time and the created particles do not y away to innity (forexample they are collected on two screens), then an electric eld is created that opposes the originaleld and at some point terminates pair production.

5.8. BOSONS IN A MAGNETIC FIELD 49

5.8 Bosons in a magnetic field

Putting a charged scalar particle in a magnetic eld is very similar to the situation of a chargedfermion: the only nontrivial dierence is the absence of spin. In addition, the spin-zero free energycomes with a relative minus sign compared to the spin-half case. Looking back at the fermionicfree energy (5.18), we only have to skip the sum over spins 𝑠𝑧, so that

𝑓 =𝑞𝐵𝑇

2𝜋∑𝑛,𝑘∫

d𝑝𝑧2𝜋

log𝑞𝐵(2𝑘 + 1) + 𝑝2𝑧 + 𝜔

2𝑛 +𝑚

2

𝑇 2. (5.43)

Referring to Fig. 5.2, the fermionic Landau levels were 𝐿𝑓𝑘 = 2𝑘𝑞𝐵 with multiplicity 2 − 𝛿𝑘,0. In

contrast, the bosonic Landau levels are at 𝐿𝑏𝑘 = (2𝑘 + 1)𝑞𝐵, with single multiplicities. Clearly, for

bosons there is no Landau level with zero energy.Using the Landau levels, we may again discover an equivalence between fermions and bosons,

just as we did at nonzero temperature. Remember there we rewrote a fermionic Matsubara-sumusing two bosonic Matsubara-sums. We needed two bosons so that the zero-mode can cancel(fermions had no zero mode). In the same spirit, we can now rewrite bosonic Landau-sums usingtwo fermionic Landau-sums. The lowest Landau-level can thus cancel (this time bosons have nolowest Landau-level). The fermionic and bosonic Landau-sums read

𝜎𝑓 = 𝑞𝐵∑𝑘

(2 − 𝛿𝑘,0)𝜙(𝐿𝑓𝑘), 𝜎𝑏 = 𝑞𝐵∑

𝑘

𝜙(𝐿𝑏𝑘) . (5.44)

Employing a very similar argument as we did for the fermion-boson equivalence at 𝑇 > 0 in (4.39),we have

𝜎𝑏(𝐵) = 𝜎𝑓(𝐵⇑2) −1

2𝜎𝑓(𝐵) . (5.45)

Since the free energy is a sum of this form, such a correspondence holds for 𝑓 as well (we just haveto watch out for the factor of −1 dierence).

Note that we cannot rewrite the fermionic sum as a combination of bosonic sums. Furthermore,note that the correspondence only holds at 𝑇 = 0. If both 𝑇 > 0 and 𝐵 > 0 are switched on, theLandau-sums are manipulated this way but the Matsubara-sums are not.

Two applications for this equivalence are the scalar QED 𝛽-function and the scalar QED pairproduction rate. For the 𝛽-function coecient, we need the quadratic part of the free energyfrom (5.23).

𝑓bos(𝒪(𝐵2),0) = −(𝑓ferm(𝐵⇑2) − 𝑓ferm(𝐵)⇑2⌋ = −

(𝑞𝐵)2

96𝜋2log

Λ2

𝑚2+(𝑞𝐵)2

48𝜋2log

Λ2

𝑚2=(𝑞𝐵)2

96𝜋2log

Λ2

𝑚2.

(5.46)Comparing to (5.26), it follows that the lowest 𝛽-function coecient for scalar QED is

𝛽1,bos =1

48𝜋2. (5.47)

It is positive, so scalar QED is also asymptotically free.For the Schwinger eect, we work with electric elds instead of magnetic elds. To go from

𝐵 to 𝐸 we needed an analytic continuation. However, this is independent of whether we workwith fermions or bosons. So the correspondence still holds. In particular, the pair production ratefrom (5.42) is

Γbos(𝐸) = −(Γferm(𝐸⇑2) − Γferm(𝐸)⇑2⌋ = −(𝑞𝐸)2

16𝜋3Li2 (𝑒

−2𝑚2𝜋⇑(𝑞𝐸)) +

(𝑞𝐸)2

8𝜋3Li2 (𝑒

−𝑚2𝜋⇑(𝑞𝐸))

= −(𝑞𝐸)2

8𝜋3Li2 (−𝑒

−𝑚2𝜋⇑(𝑞𝐸)) ,

(5.48)


where we used the relation

Li2(𝑥2) − 2Li2(𝑥) = 2Li2(−𝑥) . (5.49)

Finally, we also give the bosonic version of (5.29)

∆𝑓𝑟,bos(𝐵,𝑇,𝜇) = −1

16𝜋2∫

∞

0

d𝑠

𝑠3𝑒−𝑚

2𝑠⌊

𝑞𝐵𝑠

sinh(𝑞𝐵𝑠)Θ3 (

𝑖𝜇

2𝑇, 𝑒−1⇑(4𝑠𝑇

2)) − 1 +

(𝑞𝐵𝑠)2

6 , (5.50)

where we employed the identity

coth𝑥

2− coth𝑥 =

1

sinh𝑥, (5.51)

at 𝑇 = 0 and then generalized to 𝑇 > 0 and 𝜇 > 0 using our knowledge from Sec. 3.4

5.9 Meissner effect

We can now perform the analysis of Bose-Einstein condensation in the presence of the magneticeld. First of all we observe that the lowest possible energy is not𝑚 anymore but

⌈𝑚2 + 𝑞𝐵, due to

the fact that the lowest Landau level has nonzero energy. This implies that there is no condensationup to 𝜇𝑐 =

⌈𝑚2 + 𝑞𝐵.

Besides, there is another important dierence with respect to the 𝐵 = 0 system. Eectively, themagnetic eld reduces the spatial dimensionality to one: instead of a three-dimensional momentumintegral d3𝑝 we have 𝑞𝐵∑ℓ ∫ d𝑝𝑧, compare (2.19) to (5.18). We have seen in Sec. 3.6 that reducingthe dimensionality destroys the Bose-Einstein condensate of charged scalars. This is indeed whathappens: a lengthy calculation similar to that of Sec. 3.3.2 reveals that 𝑛therm(𝐵) in 3 + 1dimensions is innite at 𝜇 → 𝜇𝑐. Thus, the thermal part of the density can accommodate anarbitrary density, so that as we reduce the temperature at a xed density 𝑛fix, we never reach𝜇 = 𝜇𝑐 and, therefore, never produce a condensed mode.

This frustrates the system. At 𝐵 = 0 and 𝑇 < 𝑇𝑐 there is a condensate, but at any innitesimal𝐵 > 0 there isn't any. However, a small change in 𝐵 should not aect the system so drastically.The solution to the puzzle is the Meissner eect: the material expels the magnetic eld so that thecondensate can be maintained.

To investigate this in more detail, we need to distinguish between the external eld 𝐻 thatwould be present without particles and the magnetic induction 𝐵 that the charged particles feel.The latter appears in the free energy of the particles, while the former can be calculated via themagnetization

𝑀 = −𝜕𝑓

𝜕𝐵, 𝐵 =𝐻 +𝑀 . (5.52)

If the magnetic eld is small, we can do a Taylor expansion in it so that

𝑀 = 𝜒𝐵, 𝜒 =𝜕2𝑓

𝜕𝐵2⋁𝐵=0

, (5.53)

where 𝜒 is the magnetic susceptibility.

The magnetic permeability measures to what extent the two magnetic elds dier:

𝑝 =𝐵

𝐻=

1

1 − 𝜒. (5.54)

5.9. MEISSNER EFFECT 51

If ⋃𝜒⋃≪ 1, the permeability is near unity. On the other hand, to have a fully expelled eld, we need𝐵 = 0 but 𝐻 ≠ 0, so 𝜒 = −∞. Let us check this by explicit calculation.

We begin with (5.50), where we can directly dierentiate with respect to 𝐵,

𝜒 = −1

48𝜋2∫

∞

0

d𝑠

𝑠𝑒−𝑚

2𝑠]Θ3 (

𝑖𝜇

2𝑇, 𝑒−1⇑(4𝑠𝑇

2)) − 1 . (5.55)

This is an ultraviolet nite quantity: it goes to zero exponentially fast for 𝑠→ 0. By now we knowthat the Bose-Einstein condensate is an infrared phenomenon, so we need to look at the oppositeend 𝑠→∞. This region could cause 𝜒 to diverge. The mass gives an exponential decay so it seemsne, but we also need to check the Θ function in this limit,

Θ3 (𝑖𝜇

2𝑇, 𝑒−1⇑(4𝑠𝑇

2))

𝑠→∞ÐÐÐ→ 2𝑇

⌋𝜋𝑠 𝑒𝜇

2𝑠+ (exponentially suppressed⌋ , (5.56)

where the last term signals contributions of the form 𝑒−..𝑠 with negative exponent. Altogether, wethus have, concentrating on the 𝑠→∞ region,

𝜒 = −𝑇

24𝜋⌋𝜋∫

∞ d𝑠⌋𝑠𝑒(𝜇

2−𝑚2)𝑠+ (less divergent⌋ . (5.57)

The susceptibility is nite for any 𝜇 <𝑚 but diverges3 for 𝜇 =𝑚 because of the slow 1⇑⌋𝑠 decay of

the integrand. Having 𝜒 = −∞ implies through (5.54) that the magnetic permeability tends to zero the magnetic induction is zero even in the presence of a small external magnetic eld. In otherwords, the magnetic eld is expelled from the medium.

Summarizing, it is energetically favorable for the system to form a condensate (the eectivepotential for the zero mode 𝜑 is Mexican hat-like). This remains to hold if a small magneticeld is switched on. But if the charged particles are immersed in the eld, they cannot form acondensate due to the dimensional reduction and the Mermin-Wagner theorem. So the magneticeld is expelled; a condensate can form and the magnetic susceptibility is −∞.

3Remember we do the calculation via a Taylor expansion in 𝐵 at 𝐵 = 0. Thus the critical chemical potential is𝜇𝑐 =𝑚.

Appendix A

Special functions and complex analysis

In this note we discuss the denition and some properties of a few special functions that arefrequently encountered in nite temperature eld theory, see also Ref. [2]. Instead of mathematicaldetails we will concentrate on the practical applications. Check out Ref. [4] for the proper denitions.

A.1 Γ-function

The Γ-function is the generalization of the factorial,

Γ(𝑛) = (𝑛 − 1)! 𝑛 ∈N, (A.1)

to complex arguments 𝑧 ∈ C. For Re(𝑧) > 0 it is dened via the Mellin transform (see below)

Γ(𝑧) = ∫∞

0d𝑡 𝑡𝑧−1 𝑒−𝑡, (A.2)

and is analytically continued to Re(𝑧) ≤ 0 through the relation

Γ(𝑧 + 1) = 𝑧Γ(𝑧). (A.3)

The analytically continued function is meromorphic, i.e. it is holomorphic (complex dierentiable)everywhere except for a set of isolated points. For the Γ-function these isolated points are all simplepoles with residue

simple poles: 𝑧 = −𝑛, 𝑛 ∈N, Res(Γ,−𝑛) =(−1)𝑛

𝑛!. (A.4)

Fig. A.1 shows the absolute value of Γ(𝑧) for complex arguments. Note the simple poles for non-positive integers, the rapid decay towards large complex values in the left half-plane and the factorialrise in the right half-plane. Note also that the Γ-function has no zeros.

We will use the so-called duplication formula

Γ(2𝑧)

Γ(𝑧)= Γ(𝑧 + 1⇑2)22𝑧−1

1⌋𝜋, (A.5)

and the special valueΓ(1⇑2) =

⌋𝜋, (A.6)

from which values at other half-integers can also be calculated using Eq. (A.3). For example

Γ(−1⇑2) = −2⌋𝜋 . (A.7)

52

A.2. 𝜁-FUNCTION 53

Figure A.1: Magnitude of the Γ function in the complex plane.

We will also need the expansion of the Γ-function around its poles,

Γ(0+𝜖) =1

𝜖−𝛾𝐸+𝒪(𝜖), Γ(−1+𝜖) = −

1

𝜖−1+𝛾𝐸+𝒪(𝜖), Γ(−2+𝜖) =

1

2𝜖+

3

4−𝛾𝐸2+𝒪(𝜖). (A.8)

A.2 𝜁-function

The Riemann 𝜁-function is dened by the innite series or by the integral representation

𝜁(𝑧) =∞

∑𝑛=1

𝑛−𝑧 =1

Γ(𝑧)∫

∞

0d𝑥

𝑥𝑧−1

𝑒𝑥 − 1, (A.9)

which converge for Re(𝑧) > 1. An analytic continuation to Re(𝑧) > 0 can be dened via

𝜁(𝑧) =1

1 − 21−𝑧1

Γ(𝑧)∫

∞

0d𝑥

𝑥𝑧−1

𝑒𝑥 + 1, (A.10)

and to Re(𝑧) ≤ 0 through

𝜁(𝑧) = 2𝑧𝜋𝑧−1 sin(𝜋𝑧

2)Γ(1 − 𝑧)𝜁(1 − 𝑧). (A.11)

The integral in Eq. (A.10) diverges for 𝑧 = 0, but Γ(𝑧) as well, allowing for a denition of 𝜁(0) = −1⇑2.Note that this implies that in this regularization ∑∞𝑛=1 1 = −1⇑2.The analytic continuation is a meromorphic function with one simple pole at 𝑧 = 1,

simple pole: 𝑧 = 1, Res(𝜁,1) = 1, (A.12)

and zeros at each even negative integer,

𝜁(−2𝑛) = 0, 𝑛 ∈N+. (A.13)

54 APPENDIX A. SPECIAL FUNCTIONS AND COMPLEX ANALYSIS

Figure A.2: Magnitude of the 𝜁 function in the complex plane.

(There are also more zeros with nonzero imaginary part). Fig. A.2 shows the absolute value of 𝜁(𝑧)for complex arguments. Note the pole at 𝑧 = 1 and the less visible zero at 𝑧 = −2. Some specialvalues we will need are

𝜁(0) = −1

2, 𝜁(2) =

𝜋2

6, 𝜁(4) =

𝜋4

90. (A.14)

A.3 Elliptic Θ-function

A further special function that appears frequently in high-temperature calculations is the ellipticΘ-function. We will mostly use the following representation

∞

∑𝑛=−∞

𝑒−(2𝜋𝑛)2𝑠=

1

2⌋𝜋𝑠

Θ3 (0, 𝑒−1⇑(4𝑠)) , (A.15)

which is useful for scalar elds and

∞

∑𝑛=−∞

𝑒−(2𝜋(𝑛+1⇑2))2𝑠=

1

2⌋𝜋𝑠

Θ3 (𝜋

2, 𝑒−1⇑(4𝑠)) , (A.16)

which is relevant for fermions. More representations can be found Sec. 20 of [4].In both cases, in the limit 𝑠 → 0 the function approaches unity, independently of the rst

argument. The visualization of Θ3 as a function of the two variables is shown in Fig. A.3.

A.4 Mellin transform and contour integration

The Mellin transform is an integral transformation that relates two functions to each other. Wewill mostly use it for the specic case where the pair of functions is Γ(𝑧) and 𝑒−𝑡,

Γ(𝑧) = ∫∞

0d𝑡 𝑡𝑧−1 𝑒−𝑡 ↔ 𝑒−𝑡 =

1

2𝜋𝑖∫

𝑐+𝑖∞

𝑐−𝑖∞d𝑧 𝑡−𝑧 Γ(𝑧). (A.17)

A.4. MELLIN TRANSFORM AND CONTOUR INTEGRATION 55

Figure A.3: The elliptic theta function Θ3(𝑥, 𝑦).

In general, 𝑧 should take values 𝑧′ such that the integral in the rst relation exists: in our caseRe(𝑧′) > 0, since only then the 𝑡-integral is convergent for small 𝑡 (for large 𝑡 it is always convergent).Then, the inverse transform (the second relation) also holds given that 𝑐 > Re(𝑧′) in our case forpositive values of 𝑐. The inverse transform is a vertical line integral on the complex plane. For amore detailed mathematical formulation see Sec. 2.5 of Ref. [4].

We have already used the rst relation of the above pair for the denition of the Γ-function fornon-integer values in Eq. (A.2). A similar pair is → Exercise

Γ(𝑧)𝑎−𝑧 = ∫∞

0d𝑡 𝑡𝑧−1𝑒−𝑎𝑡 ↔ 𝑒−𝑎𝑡 =

1

2𝜋𝑖∫

𝑐+𝑖∞

𝑐−𝑖∞d𝑧 𝑡−𝑧 Γ(𝑧)𝑎−𝑧, (A.18)

valid for Re(𝑧) > 0 and Re(𝑎) > 0, and again 𝑐 > 0.

To calculate the inverse Mellin transform, we will need to use contour integration in the complexplane. Considering the contour 𝒞, this involves the residue of the integrand via Cauchy's residuetheorem

∮𝒞d𝑧 𝜑(𝑧) = 2𝜋𝑖∑

𝑘

Res(𝜑, 𝑧𝑘), (A.19)

where 𝑧𝑘 are the poles of the meromorphic function 𝜑, and we assumed that the curve 𝒞 windsaround the 𝑧𝑘 in the counter-clockwise direction. The pole of the function 𝜑 at 𝑧 = 𝑧𝑘 is of order 𝑛if 𝜑 behaves as

𝜑(𝑧) ∼1

(𝑧 − 𝑧𝑘)𝑛, (A.20)

in the vicinity of 𝑧𝑘. The positions and orders of the poles of a meromorphic function are unam-biguous (they are determined by the Laurent series of the function).

The residue for a simple (𝑛 = 1) pole can be calculated as

simple pole: Res(𝜑, 𝑧𝑘) = lim𝑧→𝑧𝑘

(𝑧 − 𝑧𝑘) ⋅ 𝜑(𝑧). (A.21)

Note that for a simple pole

Res(𝜑(𝑧⇑2 + 𝑐), 𝑧 = 𝑧𝑘) = 2Res(𝜑(𝑧 + 𝑐), 𝑧 = 𝑧𝑘⇑2) . (A.22)

56 APPENDIX A. SPECIAL FUNCTIONS AND COMPLEX ANALYSIS

For a pole of order 𝑛, the residue is found according to

𝑛-th order pole: Res(𝜑, 𝑧𝑘) =1

(𝑛 − 1)!lim𝑧→𝑧𝑘

𝜕𝑛−1

𝜕𝑧𝑛−1((𝑧 − 𝑧𝑘)

𝑛⋅ 𝜑(𝑧)⌋ . (A.23)

Usually we will use Eq. (A.19) to calculate integrals along the real axis or along some straightline on the complex plane. One can then `close' such a line through regions of the complex planewhere the integrand vanishes and, thus, does not contribute to the integral.

Bibliography

[1] L. Onsager. Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition.Physical Review, 65:117149, February 1944. 5

[2] M. Laine and A. Vuorinen. Basics of Thermal Field Theory. Lect. Notes Phys., 925:pp.1281,2016. 8, 32, 40, 52

[3] D. J. Toms. The Eective action at nite temperature and density with application to Bose-Einstein condensation. 1996. 17

[4] NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.13 of2016-09-16. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert,C. W. Clark, B. R. Miller and B. V. Saunders, eds. 25, 29, 52, 54, 55

[5] J.I. Kapusta and C. Gale. Finite-Temperature Field Theory: Principles and Applications. Cam-bridge Monographs on Mathematical Physics. Cambridge University Press, 2006. 40

[6] C. Itzykson and J.B. Zuber. Quantum Field Theory. Dover Books on Physics. Dover Publica-tions, 2006. 45

[7] J. S. Schwinger. On gauge invariance and vacuum polarization. Phys. Rev., 82:664679, 1951.47, 48

57

http://dlmf.nist.gov/

Documents

Thermal Quantum Field Theory - uni-frankfurt.deendrodi/thermalFT.pdf · 2020-06-02 · Quantum mechanics Let us consider the quantum mechanics of a particle of mass , moving in a