Juan’Maldacena’ - Nambu...
Transcript of Juan’Maldacena’ - Nambu...
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Models for the vacuum
Juan Maldacena
Ins4tute for Advanced Study
Nambu Memorial Symposium.
Chicago, March 11-‐13, 2016
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(BCS: 1957 )
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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.
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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 ?
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• We will describe a simple economic model that displays gauge symmetry and is found in reality in the Higgs phase.
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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
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1 current peso = 1013 pesos when I was born
Gauge symmetry in ac4on
(half of the e-‐folds of cosmic infla4on)
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Gauge poten4als = exchange rates
2 dollars = 1 euro dollar euro
Exchange rates = gauge poten4als
2,000 Ŧ = 1 euro
r = eA
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2 dollars = 1 euro
Peso
10 pesos = 1 euro
1 dollar = 6 Pesos
Euro Dollar
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2,000 Ŧ = 1 euro
Peso
10 pesos = 1 euro
1,000 Ŧ = 6 Pesos
Euro Ŧ Gauge symmetry is a local opera4on
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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
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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.
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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
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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.
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Rules
1) Countries arranged in a grid or a lakce.
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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.
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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.
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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.
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Physics is simpler than economics !
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• Pure electromagne4sm à only thing you can carry is money.
• Only exchange rates are relevant.
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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+ ✏
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1 Peso = 1 $
= 1 $
= 6 Pesos
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1 Peso = 1,000 Ŧ
= 1 ,000 Ŧ
= 6 Pesos
Gauge symmetry s4ll present. Is not broken. It is Higgsed.
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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
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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.
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• 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…
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Maxwell : Ether as a mechanical model Nambu: Ether as a superconductor. Here: Ether as a financial model.
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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.
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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
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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.
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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
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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.
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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
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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
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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.
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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.
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End of economics model
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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
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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.
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• Model is solvable in the large N limit.
• Flows to an IR almost conformal fixed point.
1
J⌧ t, � ⌧ NPower
J
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Spectrum
(specific, but random J’s)
D. Stanford
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Solvable thanks to the simple structure of diagrams
= ⌃(⌧, ⌧ 0) = J2G(⌧, ⌧ 0)3
= G(⌧, ⌧ 0) = (@⌧ � ⌃)�1
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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
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Example: Go from zero temperature to finite temperature solu4on
f(⌧) =�
⇡tan
⇡⌧
�
Gf =
"⇡
� sin ⇡⌧�
#2�
G(⌧, ⌧ 0) / 1
(⌧ � ⌧ 0)2�
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• 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
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Thermal free energy
��F = N
c1�J + s0 +#
2⇡2
�J
�
Extremal entropy Near extermal entropy à linear in T
From Nambu-‐Goldstone mechanism.
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Ground state entropy ?
⇢(E) ⇠ eNs0+Np
(E�Eg)/EgF (E)⇢(E) ⇠ eNs0F (E)
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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.
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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
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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
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ds2|Bdy =1
✏2du2
�|Bdy =1
✏�r(u)
Asympto4c boundary condi4ons:
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ds2|Bdy =1
✏2du2
ds2 =d⇢2 + sinh2 ⇢d⌧2
⇢(⌧)
1
✏2=(⇢02 + sinh2 ⇢)
✓d⌧
du
◆2
Infinite number of solu4ons.
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• Similar to the boundary gravitons of AdS3
• Here one must break the symmetry.
Turiaci Verlinde
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One one solu4on
ds2|Bdy =1
✏2du2
�|Bdy =1
✏�r(u)
Ac4on à related to Schwarzian
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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
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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
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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.
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• 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.
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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…