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Relativistic Quantum Mechanics

Dipankar Chakrabarti

Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208016, India

(Dated: August 6, 2020)

1

I. INTRODUCTION

Till now we have dealt with non-relativistic quantum mechanics. A free particle satisfying

Schrodinger equation has the non-relatistic energy E = ~p22m . Non-relativistc QM is applicable

for particles with velocity much smaller than the velocity of light(v

where gµν is the metric of the space-time. In Minkowski space

gµν =

1 0 0 0

0 −1 0 0

0 0 −1 0

0 0 0 −1

. (2)

So,

ds2 = (cdt)2 − ((dx1)2 + (dx2)2 + (dx3)2) = (cdt)2 − ( ~dr)2 (3)

Under Lorentz transformation xµ transforms as x′µ = Λµνxν where Λµ ν is a 4 × 4 matrix

representing the Lorentz transformation operator. For example, the operator for boost along

x1 axis

Λµν =

γ −γβ 0 0

−γβ γ 0 0

0 0 1 0

0 0 0 1

(4)

where β = v/c and γ = 1/ √

1− (v/c)2. So, the transformed coordinates under the boost along

x1:

ct′ = γ(ct− v c x1), x′1 = γ(x1 − v

c ct), x′2 = x2, x′3 = x3. (5)

Check that

ds′2 = (cdt′)2 − ((dx′1)2 + (dx′2)2 + (dx′3)2) = γ2(cdt− βdx′1)2 − γ2(dx1 − βcdt)2 − (dx2)2 − (dx3)2

= ds2 (6)

i.e., ds2 is Lorentz invariant. ds2 can be both positive or negative unlike spatial distance ( ~dr)2

which is always positive. If

ds2 > 0 i.e., (cdt)2 > ( ~dr)2, the interval is called "time-like"

ds2 < 0 i.e., (cdt)2 < ( ~dr)2, the interval is called "space-like"

ds2 = 0 i.e., (cdt)2 = ( ~dr)2, the interval is called "light-like".

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covariant & contravariant vectors: Any quantity which transforms like xµ under Lorentz

transformation is called a contravariant vector while anything which transforms like ∂ ∂xµ

is called

covariant vector. General convention for contravariant vector is aµ (i.e.,µ is in the superscript)

and for covariant vector aµ (i.e, µ is in the subscript) i.e, ∂∂xµ = ∂µ. The inner product of a

covariant vector and a contravariant vector is a Lorentz invariant(i.e., scalar). The contra and

covariant vectors are related by

xµ = ∑ ν

gµνx ν . (7)

Using the convention of summation over repeated indices we can write the above eqn as xµ =

gµνx ν where ν in gµν is repeated again in xν and hence is summed over. Similarly, xµ = gµνxν .

In Minkowski space, gµν = gµν . So, we have

x0 = g0νxν = g00x0 + g01x1 + g02x2 + g03x3 = g00x0 = x0 (8)

x1 = g1νxν = g10x0 + g11x1 + g12x2 + g13x3 = g11x1 = −x1. (9)

Similarly x2 = −x2 and x3 = −x3.

Inner product or scalar product of two 4-vectors is defined as

A ·B = AµBµ = (A0B0 + A1B1 + A2B2 + A3B3) = (A0B0 − A1B1 − A2B2 − A3B3) (10)

= A0B0 − ~A · ~B = gµνAµBν = gµνAµBν . (11)

Differential operators:

∂µ = ∂

∂xµ = (1

c

∂

∂t , ∂

∂x1 , ∂

∂x2 , ∂

∂x3 ) (12)

= (∂0, ∂1, ∂2, ∂3) = ( 1 c

∂

∂t , ~∇) (13)

∂µ = gµν∂ν = ( 1 c

∂

∂t ,−~∇) (14)

The Lorentz invariant second order differential operator or the d’Alembertian operator is

� = ∂µ∂µ = ( 1 c2 ∂2

∂t2 ,−( ∂

2

∂x2 + ∂

2

∂y2 + ∂

2

∂z2 )) = ( 1

c2 ∂2

∂t2 ,−∇2). (15)

We know the relativistic mass mr = γm and energy E = mrc2 = γmc2. The energy-momentum

4-vector is pµ = (E/c, ~p) where ~p = γm~v. So,

p2 = gµνpµpν = pµpµ = ( E

c )2 − (~p)2 = (γmc

2)2 c2

− (γm~v)2 = m2c2 (16)

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(in the natural unit ~ = c = 1, p2 = m2). So, the relativistic energy momentum relation is

given by E2 = (~p)2c2 +m2c4. Another useful quantity is

p · x = pµxµ = Et− ~p · ~x. (17)

For non-relativistic particle (v

scalar. Thus KG equation describes the relativistic dynamics of a scalar particle. The plane

wave solution of the KG eqn is

φ(x) = Ne−i(Et−~p·~x) (25)

where N is the normalization constant and energy E = ± √ ~p2c2 +m2c4 i.e., energy can be both

positive and negative.

Continuity Equation:

Pre-multiply Eq.(23) by φ∗(x) to get

φ∗(x) ( 1 c2 ∂2

∂t2 − ~∇2

) φ(x) = −m

2c2

~2 φ∗(x)φ(x) (26)

Now take the complex conjugate of Eq.(23) and post-multiply with φ(x), which gives

( 1 c2 ∂2

∂t2 φ∗)φ− (~∇2φ∗)φ = −m

2c2

~2 φ∗(x)φ(x) (27)

Eq(26)-Eq(27) gives:

φ∗ 1 c2 ∂2φ

∂t2 − 1 c2 ∂2φ∗

∂t2 φ− (φ∗∇2φ− φ∇2φ∗) = 0 (28)

⇒ 1 c

∂

∂t

[ i~

2mc

( φ∗ ∂φ

∂t − ∂φ

∗

∂t φ )]

+ ~∇ · [ ~ 2im

( φ∗~∇φ− (~∇φ∗)φ

)] = 0 (29)

⇒ 1 c

∂

∂t ρ+ ~∇ ·~j = 0 (30)

⇒ ∂µjµ = 0 (31)

This is the continuity equation for the KG eqn, where

j0 = ρ = i~2mc

( φ∗ ∂φ

∂t − ∂φ

∗

∂t φ )

(32)

~j = ~2im

( φ∗~∇φ− (~∇φ∗)φ

) . (33)

Recall the continuity eqn for Schrodinger equation, ρ is the probability density and ~j is the

probability current. Continuity equation has the interpretation of conservation of probability.

It tells that if the probability of finding a particle in some region decreases, the probability of

finding it out side that region increases, i.e., there is a flow of probability current so that the total

probability remains conserved. Since the KG eqn also satisfies the same continuity eqn, it is

natural to interpret ρ as the probability density and ~j as the probability current. [Note: Density

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transforms like the 0th component of a 4−vector (jµ) under Lorentz transformation. Since φ

is a Lorentz invariant quantity, φ2 does not transform like a density, but ρ defined in Eq.(32)

does.] The probability density corresponding to the plane wave solution reads ρ = 2|N |2E.

There are two major problems with the KG equation.

(1) The eqn has both positive and negative energy solutions. The negative energy solution

poses a problem! For large |~p| we can have large negative energy, i.e., the system become

unbounded from below. So, we can extract any arbitrary large amount of energy from the

system by pushing it into more and more negative energy states. One may say, we truncate

the physical space to be the positive energy states only i.e, only E = + √ ~p2c2 +m2c4 are

physical. But then (a) the eigenstates don’t form a complete basis states, (both +ve and -ve

energy states are Fourier modes of φ); if we don’t have completeness relation, we cannot have

superposition principle too ie., we cannot expand a state χ in the basis of φ ( i.e., χ = ∑i ciφi is no longer valid) and (b) a perturbation may cause the system to jump to a negative energy

states. Since -ve energy states are valid solutions of the KG equation, we can not stop that.

So, just interpreting negative energy states as unphysical does not work.

(2) The second problem is associated with the probability density. As we have seen ρ =

2|N |2E, i.e, ρ is negative if E is negative. But to interpret ρ as the probability density, it must

be positive definite.

[Though in QM, KG equation looks awkward at this moment, but in QFT this is a valid

equation for scalar (spin=0)particles. Feynman and Stückelberg interpreted the positive energy

states as particles propagating forward in time and negative energy states are propagating back-

ward in time and thus represent antiparticles propagating forward in time. But we’ll not discuss

those developments here.]

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B. Dirac Equation:

The probability density in KG eqn depends on energy and becomes negative for negative

energy. The energy in the expression of ρ appears due to the time derivative in Eq.(32). Dirac

realised that this is due to the fact that KG eqn involves second order time derivative. Notice

that Schrodinger equation invoves first order time derivative, and ρ does not involve any time

derivative.. So, if we want to write a relativistic wave equation with positive definite probability

density, the equation should be first order in time derivative. To be consistent with the Lorentz

transformations in special theory of relativity, the wave equation with first order time derivative

must also be first order in space derivatives. So, Dirac wrote the Hamiltonian as

H = α1p1c+ α2p2c+ α3p3c+ βmc2. (34)

Writing the momentum in differential operator form in the position space, we must have the

wave equation

i~ ∂ψ(x) ∂t

= ( − i~c(α1

∂

∂x1 + α2

∂

∂x2 + α3

∂