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Models for the vacuum
Juan Maldacena
Ins4tute for Advanced Study
Nambu Memorial Symposium.
Chicago, March 11-‐13, 2016
(BCS: 1957 )
Outline
• Interes4ng everyday example of a gauge theory.
• Aspects of a simple quantum mechanical model that has some features in common with near extremal black holes.
Gauge ``symmetry’’ and is realiza4ons
• Gauge ``symmetry’’ is central to modern par4cle physics.
• So is the Nambu-‐ …. -‐Higgs mechanism.
• What is a good everyday analogy for these concepts ?
• We will describe a simple economic model that displays gauge symmetry and is found in reality in the Higgs phase.
The gauge symmetry of prices
• We normally measure the price of objects in dollars.
Your salary = $ 1000 1 apple = $ 1 1 pear = $ 2
Your salary = Ŧ 1,000,000 1 apple = Ŧ 1,000 1 pear = Ŧ 2,000
1$ = 1000 Ŧ
Nothing changes à gauge symmetry ! Gauge group = R and not U(1)
Weyl, K. Illinski, K. Young, P. Malaney Observables: Price of apples
Your salary
1 current peso = 1013 pesos when I was born
Gauge symmetry in ac4on
(half of the e-‐folds of cosmic infla4on)
Gauge poten4als = exchange rates
2 dollars = 1 euro dollar euro
Exchange rates = gauge poten4als
2,000 Ŧ = 1 euro
r = eA
2 dollars = 1 euro
Peso
10 pesos = 1 euro
1 dollar = 6 Pesos
Euro Dollar
2,000 Ŧ = 1 euro
Peso
10 pesos = 1 euro
1,000 Ŧ = 6 Pesos
Euro Ŧ Gauge symmetry is a local opera4on
2 dollars = 1 euro dollar euro
Peso
10 pesos = 1 euro
6 Pesos = 1 dollar
Do you see anything interes4ng about these exchange rates ?
Opportunity to speculate = Magne4c field
2 dollars = 1 euro dollar euro
Peso
10 pesos = 1 euro
6 Pesos = 1 dollar
Speculators à move along this circuit As electrons move in circles in a magne4c field.
Electric fields
4me
UK USA
USA UK
1 % 2 %
1 $ = 1 £
1 $ = 1 £
Gauge poten4als in the 4me direc4on = interest rates
Debt money
Electric field
electron positron
4me
A financial model for the ether
• Make the model more similar to physics.
• We will make up some rules.
• These are rules that we have in physics, but are not quite true in economics.
• The rules are very simple and are the reason physics is simpler than economics.
Rules
1) Countries arranged in a grid or a lakce.
Countries are arranged in a regular palern The exchange rates can be all different. Each white link is an exchange rate. Only exchanges between neighbors.
You cannot trade by phone. You cannot fly. You can only walk from one country to the next, from there to the next and so on. Countries = points in space.
At each country you can only have the currency of that country. Imagine the white lines as a bridge. You have to exchange your money at the bridge, when you move to a new country. No commissions.
Physics is simpler than economics !
• Pure electromagne4sm à only thing you can carry is money.
• Only exchange rates are relevant.
New fields • We consider other things we can carry from one country to
the next.
• E.g. say we can carry gold from one country to the next.
• Gold has a price at each country: p(x)
• Under a local gauge transforma4on p(x) changes.
r = eA , p = ec ,
A ! A+ d✏ , c ! c+ ✏
1 Peso = 1 $
= 1 $
= 6 Pesos
1 Peso = 1,000 Ŧ
= 1 ,000 Ŧ
= 6 Pesos
Gauge symmetry s4ll present. Is not broken. It is Higgsed.
1 Peso = 1 $
= 1 $
= 6 Pesos
New opportuni4es to speculate!
money gold
What do we call this in physics ?
gain =P (~x)ri(~x)
P (~x+ ei)⇠ 1� [c(~x+ ei)� c(~x)�Ai(~x)] = 1�Dic
1 P = 6 $
= 1 $
= 1 P
Choose the currencies so that the price of gold is one à Unitary gauge. The exchange rates remain as variables. The opportuni4es to speculate are clearer now. The fact that there is now a special exchange rate is related to the mass genera4on, the mass of the gauge bosons.
• What we described so far is just the kinema4cs of gauge theory.
• We can ask whether we can recover the dynamics with decent assump4ons.
• Not any different from Maxwell’s model…
Maxwell : Ether as a mechanical model Nambu: Ether as a superconductor. Here: Ether as a financial model.
Gekng Maxwell’s equa4ons
• Two versions. First the Euclidean equa4ons. • Then the Lorentzian equa4ons.
• We assume small devia4ons from one to one rates, and small devia4ons of prices from one. And work to first order in these devia4ons.
Short range speculators • Assume the existence of speculators that follow the simplest elementary plaquets in the lakce.
• If there is an imbalance, they start circula4ng carrying an amount of money propor4onal to the imbalance, propor4onal to the magne4c field.
m = A1(~x) +A2(~x+ 1)�A1(~x+ 2)�A2(~x) = F12(~x)
12
~x
~x+ 1
~x+ 2
Gekng Maxwell’s equa4ons
• Demand that the net flow of money along any link is zero.
• With no gold…
X
i
@iFij = 0
If we include 4me as before, we s4ll get the euclidean equa4ons.
Massive vector equa4ons • Demand that the net flow of money along any link is zero.
• With gold… • Also “gold circuit speculators” moving along each link, carrying an amount of money propor4onal to the gain percentage.
X
i
@iDic = 0
No net money flux at links
No net gold accumula4on at countries
X
i
@iFij +Dic = 0
Lorentzian vector field equa4ons
• Now assume that there is 4me. • Simple unrealis4c op4on à assume that speculators along circuits involving the 4me direc4on want to lose money à get the extra minus sign.
• A beler op4on is the following.
Lorentzian equa4ons.
• As before, assume that speculators move along spacelike circles, carrying an amount of money that is propor4onal to the gain.
• Assume that the banks change the exchange rate with a speed that is propor4onal to the total imbalance of currencies accumulated. (There is some iner4a)
(Asssume A0 = 0 gauge).
Aj =
Z t X
i
@iFij Aj =X
j
@iFij
With gold
• Similar dynamics for the change of price of gold.
• Correct Proca equa4ons.
�@0D0c+ @iDic = 0
Fine tuned the coefficients to make the speed of the longitudinal and transverse modes the same.
�Ai + @jFji +Dic = 0
Comments
• Importance of the spa4al arrangement of countries à structure of space.
• Short distance speculators à massive fields we integrate out and give rise to the kine4c terms of gauge fields. Emergent kine4c terms for the gauge fields.
Real economy
• Everything interac4ng with every other. But speculators only exploi4ng circuits that involve a few variables at a 4me.
• In some cases this can lead to interes4ng behavior, as we will see in a different context later.
End of economics model
Models with random interac4ons
I will talk about a par4cular model which had condensed maler roots but is interes4ng for the gauge/gravity duality.
Sachdev, Yee, Kitaev Georges, Parcollet
Polchinski, Rosenhaus, Anninos, Anous, Denef
Douglas Stanford & JM, to appear
Sachdev, Yee, Kitaev model
H =X
i1,··· ,i4
Ji1i2i3i4 i1 i2 i3 i4
Js à either random or slowly varying
{ i, j} = �ij N Majorana fermions or Gamma matrices.
Quantum mechanical model, only 4me.
N fermions , N large
hJ2i1i2i3i4i = J2/N3
J = single dimension one coupling.
• Model is solvable in the large N limit.
• Flows to an IR almost conformal fixed point.
1
J⌧ t, � ⌧ NPower
J
Spectrum
(specific, but random J’s)
D. Stanford
Solvable thanks to the simple structure of diagrams
= ⌃(⌧, ⌧ 0) = J2G(⌧, ⌧ 0)3
= G(⌧, ⌧ 0) = (@⌧ � ⌃)�1
In the IR à Conformal symmetry
G = (@⌧ � ⌃)�1 �! G ⇤ ⌃ = 1
⌃(⌧, ⌧ 0) = J2G(⌧, ⌧ 0)3
If G is a solu4on, and we are given an arbitrary func4on f(τ), we can generate another solu4on:
G �! Gf (⌧, ⌧0) = [f 0(⌧)f 0(⌧ 0)]�G(f(⌧), f(⌧ 0))
G(⌧, ⌧ 0) / 1
(⌧ � ⌧ 0)2�Is a solu4on
Example: Go from zero temperature to finite temperature solu4on
f(⌧) =�
⇡tan
⇡⌧
�
Gf =
"⇡
� sin ⇡⌧�
#2�
G(⌧, ⌧ 0) / 1
(⌧ � ⌧ 0)2�
• Is nice! • Problem à Infinite number of solu4ons. • f à like a Nambu-‐Goldstone boson.
• Fix: Remember that the symmetry is also explicitly broken (like the pion mass).
S = �N#
J
Zdt Sch(f, t) , Sch(f, t) =
✓f 00
f 0
◆0� 1
2
f 002
f 02
Thermal free energy
��F = N
c1�J + s0 +#
2⇡2
�J
�
Extremal entropy Near extermal entropy à linear in T
From Nambu-‐Goldstone mechanism.
Ground state entropy ?
⇢(E) ⇠ eNs0+Np
(E�Eg)/EgF (E)⇢(E) ⇠ eNs0F (E)
4 point func4on
• We expected a conformal invariant answer. • But, due to the reparametriza4on zero modes à infinity.
• Adding the Nambu-‐Goldstone (euclidean) ac4on à get a finite answer. But is not conformal.
• S4ll the conformal symmetry and its slight breaking are running the show!
• a-‐CFT = a-‐CFT1 = how conformal symmetry is realized in QM.
a-‐AdS2/a-‐CFT1
• Gravity in AdS2 does not make sense, when we add finite energy excita4ons.
• Slightly break the symmetry. • Simplest model: Teitelboim Jackiw
Almheiri Polchinski
Ground state entropy
Comes from the volume of the addi4onal dimensions, if we are gekng this from 4 d gravity for a near extremal black hole.
Zd
2x
pg�(R+ 2) + �0
Zd
2x
pgR
Zpg�(R+ 2)
Equa4on of mo4on for φ à metric is AdS2 Equa4on of mo4on for the metric à phi is almost completely fixed
ds2 = d⇢2 + sinh2 ⇢d⌧2
� = �h cosh ⇢ Value at the horizon Posi4on of the horizon.
In the full theory: when φ is sufficiently large à change to a new UV theory
ds2|Bdy =1
✏2du2
�|Bdy =1
✏�r(u)
Asympto4c boundary condi4ons:
ds2|Bdy =1
✏2du2
ds2 =d⇢2 + sinh2 ⇢d⌧2
⇢(⌧)
1
✏2=(⇢02 + sinh2 ⇢)
✓d⌧
du
◆2
Infinite number of solu4ons.
• Similar to the boundary gravitons of AdS3
• Here one must break the symmetry.
Turiaci Verlinde
One one solu4on
ds2|Bdy =1
✏2du2
�|Bdy =1
✏�r(u)
Ac4on à related to Schwarzian
S =
Zd
2x
pg�(R+ 2)� 2
Z�r(u)
✏
2duK !
S =1
✏
2�Z
du�r(u)Sch(t, u)
t(u) t = Usual AdS2 4me coordinate u = Boundary system (quantum mechanical) 4me coordinate
Proper4es fixed by the Schwarzian • Free energy • Part of the four point func4on that comes from the explicit conformal symmetry breaking. This part leads to a chaos-‐like behavior with maximal growth in the commutator.
• Both agree with the a-‐AdS2 problem.
• We have done more computa4ons that depend on the details of the model and can be thought of as coming from addi4onal fields in a-‐AdS2 . We found the spectrum and computed the J-‐independent parts of the four point func4on.
See Shenker’s talk
growth of commutators ⇠ 1
N(�J)e2⇡t/� Kitaev
The bulk
G(⌧, ⌧ 0) �! t =⌧ + ⌧ 0
2, � =
⌧ � ⌧ 0
2
Conformal Casimir on G à Wave operator in AdS2
Map : (2 point on the boundary ) à one point in the bulk.
• This no4on of a-‐AdS2 is similar to:
• Infla4on = a-‐dS = almost de-‐Siler. We need a scalar field to have infla4on end and to lead to the observable universe.
• In fact a-‐dS2 is a type of infla4onary theory, except that the inflaton is not a dynamical field.
Conclusions
• The idea of the vacuum as a superconductor is correct in many ways.
• 1) The Standard Model.
• 2) Black holes are like a high Tc superconductor (strongly interac4ng).
• 3) Even in the real economy…
• 4) More but 4me is too short to describe…
