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Electric fields and quantum wormholes

Engelhardt, D.; Freivogel, B.W.; Iqbal, N.

Published in:Physical Review D. Particles, Fields, Gravitation, and Cosmology

DOI:10.1103/PhysRevD.92.064050

Link to publication

Citation for published version (APA):Engelhardt, D., Freivogel, B., & Iqbal, N. (2015). Electric fields and quantum wormholes. Physical Review D.Particles, Fields, Gravitation, and Cosmology, 92(6), [064050]. DOI: 10.1103/PhysRevD.92.064050

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Download date: 17 Jul 2018

Electric fields and quantum wormholes

Dalit Engelhardt,1,2,* Ben Freivogel,2,3,† and Nabil Iqbal2,‡1Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA

2Institute for Theoretical Physics, University of Amsterdam, Science Park 904,Postbus 94485, 1090 GL Amsterdam, The Netherlands

3GRAPPA, University of Amsterdam, Science Park 904, Postbus 94485,1090 GL Amsterdam, The Netherlands

(Received 28 May 2015; published 29 September 2015)

Electric fields can thread a classical Einstein-Rosen bridge. Maldacena and Susskind have recentlysuggested that in a theory of dynamical gravity the entanglement of ordinary perturbative quanta should beviewed as creating a quantum version of an Einstein-Rosen bridge between the particles, or a “quantumwormhole.” We demonstrate within low-energy effective field theory that there is a precise sense in whichelectric fields can also thread such quantum wormholes. We define a nonperturbative “wormholesusceptibility” that measures the ease of passing an electric field through any sort of wormhole. Thesusceptibility of a quantumwormhole is suppressed by powers of theUð1Þ gauge coupling relative to that for aclassical wormhole but can be made numerically equal with a sufficiently large amount of entangled matter.

DOI: 10.1103/PhysRevD.92.064050 PACS numbers: 04.70.Dy, 04.62.+v, 11.10.Wx

I. INTRODUCTION

The past decade has seen accumulating evidence of adeep connection between classical spacetime geometry andthe entanglement of quantum fields. In the AdS/CFTcontext, there appears to be a precise holographic sensein which a classical geometry is “emergent” from quantumentanglement in a dual field theory (see e.g. [1–10]).Recently, Maldacena and Susskind have made a stronger

statement: that the link between entanglement and geom-etry exists even without any holographic changes of dualityframe [11]. They propose that any entangled perturbativequantum matter in the bulk of a dynamical theory ofgravity, such as an entangled Einstein-Podolsky-Rosen(EPR) [12] pair of electrons, is connected by a “quantumwormhole,” or some sort of Planckian, highly fluctuating,version of the classical Einstein-Rosen (ER) [13] bridgethat connects the two sides of an eternal black hole.Notably, while it clearly resonates well with holographicideas [14–29], this “ER ¼ EPR” proposal is more generalin that it makes no reference to gauge-gravity duality. Theentangled quantum fields here exist already in a theory ofdynamical gravity rather than in a holographically dualfield theory.It is not at all obvious that quantum wormholes so

defined—i.e. just ordinary entangled perturbative matter—exhibit properties similar to those of classical wormholes.For example, if we have dynamical electromagnetism, thenthe existence of a smooth geometry in the throat of anEinstein-Rosen bridge means that there exist states with a

continuously tunable1 electric flux threading the wormhole,as shown in Fig. 1. Wheeler has described such states as“charge without charge” [31].On the other hand, the two ends of a quantum wormhole

may be entangled but are not connected by a smoothgeometry. One might naively expect that Gauss’s lawwould then preclude the existence of states with a con-tinuously tunable electric flux through the wormhole. Themain point of this paper is to demonstrate that this intuitionis misleading: we will show that a quantum wormhole,made up of only entangled (and charged) perturbativematter, also permits electric fields to thread it in a mannerthat to distant observers with access to information aboutboth sides appears qualitatively the same as that for aclassical ER bridge. As we will show, for this equivalenceto hold, it is crucial that there exist sufficiently lightperturbative matter that is charged under the Uð1Þ gaugefield. It is interesting to note that similar constraints on thecharge spectrum in theories of quantum gravity have beenconjectured on different grounds [32].To quantify this, for any state jψi of either system we

define a dimensionless quantity called the “wormholeelectric susceptibility” χΔ,

χΔ ≡ hψ jΦ2Δjψi; ð1:1Þ

*engelhardt@physics.ucla.edu†benfreivogel@gmail.com‡n.iqbal@uva.nl

1Note that here and throughout the rest of the paper, the word“tunable”means only that there exists a family of states where theexpectation value of the flux is continuously tunable. The fluxcannot actually be tuned by any local observer, as the two sides ofthe wormhole are causally disconnected. There does exist aquantum tunneling process in which such flux-threaded blackholes can be created from the vacuum in the presence of a strongelectric field [30].

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with ΦΔ the electric flux through the wormhole. Thisquantity clearly measures fluctuations of the flux, and weshow below that through linear response it also determinesthe flux obtained when a potential difference is appliedacross the wormhole. This susceptibility is a particularmeasure of electric field correlations across the two sidesthat can be interpreted as measuring how easily an electricfield can penetrate the wormhole. We note that it is a globalquantity that requires knowledge of the entire state: as weshow explicitly below, measuring the wormhole suscep-tibility requires access to information about the flux on bothsides, and therefore no information is transmitted across thewormhole with this electric field.In Sec. II we compute this susceptibility for a classical

ER bridge and in Sec. III for EPR entangled matter andcompare the results. In Sec. IV we discuss how one mightpass a Wilson line through a quantum wormhole. In Sec. Vwe discuss what conditions the quantum wormhole shouldsatisfy for its throat to satisfy Gauss’s law for electric fieldsand concludewith some implications and generalizations ofthese findings.Our results do not depend on a holographic description

and rely purely on considerations from field theory andsemiclassical relativity.

II. CLASSICAL EINSTEIN-ROSEN BRIDGE

We first seek a precise understanding of what it means tohave a continuously tunable electric flux through a classicalwormhole. We begin with the action

S ¼Z

d4xffiffiffiffiffiffi−g

p �1

16πGNR −

1

4g2FF2

�: ð2:1Þ

where gF denotes the Uð1Þ gauge coupling. On generalgrounds we expect that in any theory of quantum gravity alllow-energy gauge symmetries, including the Uð1Þ above,should be compact [32,33]. This implies that the specifi-cation of the theory requires another parameter: the mini-mum quantum of electric charge, q. Throughout this paperwe will actually work on a fixed background, not allowing

matter to back-react: thus we are working in thelimit2 GN → 0.

A. Wormhole electric susceptibility

This action admits the eternal Schwarzschild black holeas a classical solution. It has two horizons which wehenceforth distinguish by calling one of them “left” and theother “right”. They are connected in the interior by anEinstein-Rosen bridge [13]. On each side the metric is

ds2¼−�1−

rhr

�dt2þ dr2

ð1− rhr Þþ r2dΩ2 r> rh; ð2:2Þ

and at t ¼ 0 the two sides join at the bifurcation sphere atr ¼ rh. The inverse temperature of the black hole is givenby β ¼ 4πrh.We now surround each horizon with a spherical shell of

(coordinate) radius a > rh. Consider the net electric fieldflux through each of these spheres:

ΦL;R ≡ 1

g2F

ZS2d~A · ~EL;R; ð2:3Þ

where the orientation for the electric field on the left andright sides is shown in Fig. 2.There is an important distinction between the total flux

and the difference in fluxes, defined as

ΦΣ ≡ ΦR þ ΦL ΦΔ ≡ 1

2ðΦR − ΦLÞ: ð2:4Þ

Via Gauss’s law, the total flux ΦΣ simply counts the totalnumber of charged particles inside the Einstein-Rosenbridge. It is “difficult” to change, in that changing itactually requires the addition of charged matter to the

FIG. 2. Electric fluxes for Einstein-Rosen bridge. Black arrowsindicate sign convention chosen for fluxes. Electric field linesthread the wormhole, changing the value of ΦΔ ≡ ðΦR − ΦLÞ=2,when a potential difference is applied across the two sides.

FIG. 1 (color online). Left: a classical Einstein-Rosen bridgewith an applied potential difference has a tunable electric fieldthreading it. Right: A “quantum wormhole,”—i.e. chargedperturbative matter prepared in an entangled state, with noexplicit geometric connection between the two sides also has aqualitatively similar electric field threading it.

2If we studied finite GN , allowing for the back-reaction of theelectric field on the geometry, then at the nonlinear level in μ wewould find instead the two-sided Reissner-Nordstrom black hole.For the purposes of linear response about μ ¼ 0 this reduces tothe Schwarzschild solution studied here.

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action (2.1). Furthermore it will always be quantized inunits of the fundamental electric charge q.On the other hand, ΦΔ measures instead the electric field

through the wormhole. It appears that it can be continu-ously tuned.We present a short semiclassical computation to dem-

onstrate what we mean by this. We set up a potentialdifference V ¼ 2μ between the left and right spheres byimposing the boundary conditions AtðrR ¼ aÞ ¼ μ,AtðrL ¼ aÞ ¼ −μ. This is a capacitor with the two platesconnected by an Einstein-Rosen bridge. The resultingelectric field in this configuration can be computed bysolving Maxwell’s equations, which are very simple interms of the conserved flux

Φ ¼ 1

g2F

Zd2Ω2r2Frt ∂rΦ ¼ 0: ð2:5Þ

As there is no charged matter all the different fluxes areequivalent: ΦR ¼ −ΦL ¼ ΦΔ. By symmetry we haveAtðrhÞ ¼ 0, and so we have

μ ¼ AtðrR ¼ aÞ ¼Z

a

rh

drFrt ¼ Φg2F4π

�1

a−

1

rh

�: ð2:6Þ

We now take a → ∞ for simplicity to find:

ΦΔ ¼�4πrhg2F

�μ ð2:7Þ

As we tune the parameter μ, ΦΔ changes continuously aswe pump more electric flux through the wormhole. There isthus a clear qualitative difference between ΦΔ and thequantized total flux ΦΣ. In this system the difference aroseentirely from the fact that there is a geometric connectionbetween the two sides.We now seek a quantitative measure of the strength of

this connection. The prefactor relating the flux to smallfluctuations of μ away from zero in (2.7) is a goodcandidate. To understand this better, we turn now to thefull quantum theory ofUð1Þ electromagnetism on the blackhole background. The prefactor is actually measuring thefluctuations of the fluxΦΔ around the μ ¼ 0 ground state ofthe system and is equivalent to the wormhole electricsusceptibility defined in (1.1):

χΔ ≡ hψ jΦ2Δjψi: ð2:8Þ

To see this, recall that we are studying the Hartle-Hawkingstate for the Maxwell field. When decomposed into twohalves this state takes the thermofield double form [1,34]:

jψi≡ 1ffiffiffiffiZ

pXn

jn�iLjniR exp�−β

2En

�: ð2:9Þ

Here L and R denote the division of the Cauchy slice att ¼ 0 into the left and right sides of the bridge, n labels theexact energy eigenstates of the Maxwell field, En denotesthe energy with respect to Schwarzschild time t, and jn�i isthe CPT conjugate of jni.3The two-sided black hole has a nontrivial bifurcation

sphere S2. The electric flux through this S2 is a quantumdegree of freedom that can fluctuate. In the decompositionabove we have two separate operators ΦL;R, both of whichare conserved charges with discrete spectra, quantized inunits of q: Φ ¼ qZ. Each energy eigenstate can be pickedto have a definite flux Φn: jn;Φni. Importantly, CPTpreserves the energy but flips the sign of the flux.Schematically, we have:

CPTjn;Φni ¼ jn;−Φni: ð2:10Þ

This means that each L state in the sum (2.9) is paired withan R state of opposite flux, and so the state is annihilated byΦL þ ΦR:

ðΦL þ ΦRÞjψi ¼ 0 ð2:11Þ

This relation is Gauss’s law: every field line entering theleft must emerge from the right.On the other hand, ΦΔ does not have a definite value on

this state: as ΦΔ does not annihilate jψi, the wave functionhas a spread centered about zero, as shown in Fig. 3. Thespread of this wave function is measured by the wormholesusceptibility (2.8). The intuitive difference between ΦΣand ΦΔ discussed above can be traced back to the fact thatthe wave function is localized in the former case andextended in the latter.The location of this maximum is not quantized and can

be continuously tuned. For example, let us deform (2.9)with a chemical potential μ:

jψðμÞi≡ 1ffiffiffiffiZ

pXn

jn�iLjniR exp�−β

2ðEn − μΦnÞ

�: ð2:12Þ

FIG. 3. Wave function of Maxwell state as a function of discretefluxes ΦΔ and ΦΣ. The spread in ΦΔ is measured by thewormhole susceptibility χΔ. The wave function has no spreadin ΦΣ.

3The pairing of a state jni with its CPT conjugate jn⋆i can beunderstood as following from path-integral constructions of thethermofield state by evolution in Euclidean time.

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Similarly deformed thermofield states and the existence offlux through the Einstein-Rosen bridge have been recentlystudied in [35–37]. Expanding this expression to linearorder in μ we conclude that

hψðμÞjΦΔjψðμÞi ¼ βμχΔ þ � � � ð2:13Þ

with χΔ the wormhole susceptibility (2.8) evaluated on theundeformed state (2.9).4 This expectation value is theprecise statement of what was computed semiclassicallyin (2.7)5: comparing these two relations we see that thewormhole susceptibility for the black hole is

χERΔ ¼ 1

g2F: ð2:15Þ

B. Quantization of flux sector on black hole background

It is instructive to provide a more explicit derivation of(2.15) by computing the full wave function as a function ofΦΔ. This requires the determination of the energy levels Enin (2.12). We study the free Maxwell theory on a fixedbackground, neglecting gravitational backreaction. As weare interested in the total flux, we need only determine aneffective Hamiltonian describing the quantummechanics ofthe flux sector. We ignore fluctuations in Aθ;ϕ and anyangular dependence of the fields, integrating over the S2 in(2.1) to obtain the reduced action:

S ¼ −2π

g2F

Zdrdt

ffiffiffiffiffiffi−g

pgrrgttðFrtÞ2: ð2:16Þ

To compute the En we pass to a Hamiltonian formalismwith respect to Schwarzschild time t. We first consider theHilbert space of the right side of the thermofield doublestate (2.9). The canonical momentum conjugate to Ar is theelectric flux:

Φ≡ δLδ∂tAr

¼ 4π

g2F

ffiffiffiffiffiffi−g

pFrt: ð2:17Þ

At does not have a conjugate momentum. The Hamiltonianis constructed in the usual way as H ≡ Φ∂tAr − L and is

H ¼Z

∞

rh

dr

�−

g2F8π

ffiffiffiffiffiffi−gpgrrgtt

Φ2 − ð∂rΦÞAt

�

þ ΦðAtðrhÞ − Atð∞ÞÞ; ð2:18Þ

where we have integrated by parts. The equation of motionfor At is Gauss’s law, setting the flux to a con-stant: ∂rΦ ¼ 0.There are two boundary terms of different character. The

value of Atð∞Þ≡ μ at infinity is set by boundary con-ditions. If μ is nonzero, then the Hamiltonian is deformed tohave a chemical potential for the flux as in (2.12). Recall,however, that the susceptibility is defined in the unde-formed state, as in (1.1). For the remainder of this sectionwe therefore set μ to zero. On the other hand, AtðrhÞ is adynamical degree of freedom. We should thus combine thisHamiltonian with the corresponding one for the left side ofthe thermofield state; demanding that the variation of thehorizon boundary term with respect to AtðrhÞ vanishes thenrequires that ΦL ¼ −ΦR, as expected from (2.11).As the flux is constant we may now perform the integral

over r to obtain the very simple Hamiltonian,

H¼GΦ2 G≡−Z

∞

rh

drg2F

8πffiffiffiffiffiffi−gp

grrgtt¼ g2F8πrh

: ð2:19Þ

This Hamiltonian describes the energy cost of fluctuationsof the electric field through the horizon of the black hole.The flux operator in the reduced Hilbert space6 of the

flux sector acts as

Φjmi ¼ qmjmi m ∈ Z; ð2:20Þ

wherem ∈ Z denotes the number of units of flux carried byeach state jmi. The Hamiltonian (2.19) is diagonal in thisflux basis, with the energy of a state with m units of fluxgiven by

Em ¼ g2F8πrh

ðqmÞ2 m ∈ Z: ð2:21Þ

Though we do not actually need it, for completeness wenote that the operator that changes the value of the fluxthrough the S2 is a spacelike Wilson line that pierces itcarrying charge q:

W ¼ exp

�iq

ZdrAr

�: ð2:22Þ

Indeed from the fundamental commutation relation½ArðrÞ;Φðr0Þ� ¼ iδðr − r0Þ we find the commutator

4The discussion in the bulk of the text assumes that we workonly to linear order in μ. If we relax this assumption then (2.13) isreplaced by

ddμ

hΦΔiμ ¼ βðhΦ2Δi − hΦΔi2Þμ; ð2:14Þ

where the right-hand side is now the appropriate generalization ofthe wormhole electric susceptibility to nonzero μ, reducing to(2.8) in the limit that μ → 0.

5More precisely: the classical relation (2.7) amounts to asaddle-point evaluation of a particular functional integral whichevaluates expectation values of (2.9).

6Where, as above, we neglect fluctuations along the angulardirections.

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½W;Φ� ¼ −qW; ð2:23Þ

meaning that a Wilson line that pierces the S2 onceincreases the flux by q. In our case any Wilson line thatpierces the left sphere must continue to pierce the right:thus if it increases the left flux it will decrease the right flux,and we are restricted to the gauge-invariant subspace that isannihilated by ΦL þ ΦR.Thus we see that the thermofield state (2.9) in the flux

sector takes the simple form

jψi ¼ 1ffiffiffiffiZ

pXm∈Z

j −mijmi exp�−g2F4ðqmÞ2

�; ð2:24Þ

where we have used β ¼ 4πrh. Since ΦΔ ¼ ΦR on thisstate, the probability of finding any flux ΦΔ through theblack hole is simply

PðΦΔÞ ¼1

Zexp

�−g2F2Φ2

Δ

�: ð2:25Þ

This is the (square of the) wave function shown in Fig. 3:even though ΦΔ has a discrete spectrum, the wave functionis extended in ΦΔ. In the semiclassical limit ðgFqÞ → 0 thediscreteness of ΦΔ can be ignored and the spread χΔ ¼hΦ2

Δi is again χERΔ ¼ g−2F , in agreement with the result foundfrom the classical analysis (2.15). This result is exact onlyfor the free Maxwell theory on a fixed background: if westudy an interacting theory (e.g. by including gravitationalbackreaction or charged matter) then the wave function willno longer be a pure Gaussian and (2.25) will receivenonlinear corrections in ΦΔ.The probability distribution exhibited in (2.25) may be

surprising as it shows that an observer hovering outside anuncharged eternal black hole nevertheless finds a nonzeroprobability of measuring an electric flux through thehorizon. However, due to (2.11) the flux measured bythe right observer will always be precisely anticorrelatedwith that measured by the left observer. These observers aremeasuring fluctuations of the field through the wormhole,not fluctuations of the number of charges inside. Through(2.13) we see that it is actually the presence of thesefluctuations that makes it possible to tune the electric fieldthrough the wormhole. In the above analysis, we havecomputed the fluctuations in the Hartle-Hawking state;more generally, any nonsingular state of the gauge fields inthe ER background will have correlated fluctuations in theflux, arising from the correlated electric fields near thehorizon.

III. QUANTUM WORMHOLE

We now consider the case of charged matter in anentangled state but with no geometrical, and hence nogravitational, connection. We will show below that when

we apply a potential difference, an appropriate pattern ofentanglement between the boxes is sufficient to generate anonvanishing electric field even though the two boxes arecompletely noninteracting.The configuration that we study here is that of a complex

scalar field ϕ charged under a Uð1Þ symmetry (withelementary charge q), confined to two disconnected spheri-cal boxes of radius a, as shown in Fig. 4. The confinementto r < a is implemented by imposing Dirichlet boundaryconditions for the fields. These boundary conditions stillallow the radial electric field to be nonzero at the boundary,so our main observable, the electric flux, is not constrainedby the boundary conditions.The action in each box is

S ¼Z

d4xffiffiffiffiffiffi−g

p �−jDϕj2 −m2ϕ†ϕ −

1

4g2FF2

�; ð3:1Þ

where Dμϕ ¼ ∂μϕ − iqAμϕ. This is now an interactingtheory where the perturbative expansion is controlledby ðgFqÞ2.Gauss’s law relates the electric flux to the total global

chargeQ. Thus we have the following operator equation onphysical states:

Φ ¼ 1

g2F

Zd3xð∇ · EÞ ¼ Q: ð3:2Þ

At first glance this situation is rather different from theclassical black hole case. ΦL and ΦR simply measure thenumber of particles in the left and right boxes respectively.There appears to be no difference between ΦΣ and ΦΔ andthus no way to thread an electric field through the boxes.This intuition is true in the vacuum of the field theory,which is annihilated by bothΦL andΦR. As it turns out, it iswrong in an entangled state.Let us now perform the same experiment as for the black

hole: we will set up a potential difference of 2μ between thetwo spheres by studying the analog of the deformedthermofield state (2.12). We will work at weak coupling:

FIG. 4. Setup for quantum wormhole. The two boxes aregeometrically disconnected but contain a scalar field in anentangled state. Correlated charge fluctuations effectively allowelectric field lines to travel from one box to the other when apotential difference is applied.

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the only effect of the nonzero coupling is to relate the fluxto the global charge as in (3.2). We are thus actuallystudying charge fluctuations of the scalar field. Thesecharge fluctuations source electric fields which cost energy,but this energetic penalty can be neglected at lowest orderin the coupling.7

The full state for the combined Maxwell-scalar system isformally the same as (2.12). We schematically label thescalar field states by their energy and global charge asjn;Qni. Due to the constraint (3.2), the scalar field sector ofthe thermofield state can be written

jψðμÞi≡ 1ffiffiffiffiZ

pXn

jn;−QniLjn;QniR exp�−β

2ðEn−μQnÞ

�:

ð3:3Þ

This state corresponds to having a constant value of At ¼ μin the right sphere and At ¼ −μ in the left sphere. Note thatwe have

ðΦL þ ΦRÞjψðμÞi ¼ ðQL þQRÞjψðμÞi ¼ 0: ð3:4Þ

We now seek to compute hΦΔiμ ¼ hQΔiμ ¼ hQRiμ wherethe second equality follows from (3.4). However tocompute QRðμÞ we can trace out the left side. Tracingout one side of the thermofield state results in a thermaldensity matrix for the remaining side: thus we are simplyperforming a standard statistical mechanical computationof the charge at finite temperature and chemical potential.Details of this computation are in the Appendix, and theresult is

hΦΔiμ ¼ q2Xn

�1

1 − coshðβωnÞ�ðβμÞ þOðμ2Þ; ð3:5Þ

where the ωn are the single-particle energy levels. The sumcan be done numerically.We conclude that the wormhole susceptibility for this

state is

χEPRΔ ¼ q2fðmβ; maÞ; ð3:6Þ

with f a calculable dimensionless function that is Oð1Þ inthe couplings and is displayed for illustrative purposes inFig. 5. Crudely speaking it measures the number ofaccessible charged states. If we decrease the entanglementby lowering the temperature, the susceptibility vanishesexponentially as f ∼ expð−ω0βÞ, with ω0 the lowest single-particle energy level. Its precise form—beyond the fact thatit is nonzero in the entangled state—is not important for ourpurposes.

We see then that as a result of the potential difference thatwe have set up between the two entangled spheres, we areable to measure flux fluctuations across the wormhole thatare both continuously tunable and fully correlated witheach other: this is observationally indistinct from measur-ing an electric field through the wormhole. This field existsnot because of geometry, but rather because the electricfield entering one sphere attempts to create a negativecharge. Due to the entanglement this results in the creationof a positive charge in the other sphere, so that the resultingfield on that side is the same electric field as the oneentering the first sphere. This appears rather different fromthe mechanism at play for a geometric wormhole, but thekey fact is that the wave function in the flux basis takesqualitatively the same form (i.e. that shown in Fig. 3) forboth classical and quantum wormholes, meaning that theuniversal response to electric fields is the same for bothsystems.Quantitatively, however, there is an important difference.

If the function f that measures the number of charged statesis of Oð1Þ, then the wormhole susceptibility for thequantum wormhole (3.6) is smaller than that for theclassical wormhole (2.15) by a factor of ðgFqÞ2. It is muchharder—i.e. suppressed by factors of ℏ—to push an electricfield through a quantum wormhole. Alternatively, we canview (2.15) as defining the value of the Uð1Þ gaugecoupling in the wormhole region. In the quantumwormholewe have succeeded in creating a putative region throughwhich a Uð1Þ gauge field can propagate, but its couplingthere (as measured by (3.6)) is large, and becomes larger asthe entanglement is decreased. Notwithstanding theselarge “quantum fluctuations,” the quantum wormholedoes nevertheless satisfy topological constraints such asGauss’s law.Despite this suppression, there is no obstruction in

principle to making f sufficiently large so that the sus-ceptibilities can be made the same. Increasing the temper-ature or the size of the box will increase the number of

1 2 3 4 5m

15

10

5

5

10

15Log f m , ma

FIG. 5. Numerical evaluation of logarithm of dimensionlessfunction fðmβ; maÞ appearing in wormhole susceptibility forcomplex scalar field. From bottom moving upwards, three curvescorrespond to ma ¼ 1; 1.5; 2. Dashed line shows asymptoticbehavior for ma ¼ 1 of exp ð−ω0βÞ.

7It is interesting to note that in the black hole case the keydifference is that the energy cost associated to the gauge fields—which we neglect in this case—is the leading effect.

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charged states and thus increase f, as can be seen explicitlyin Fig. 5. Thus even the numerical value of the EPRwormhole susceptibility can be made equal or even greaterthan that of the ER bridge, although we will require a largenumber of charged particles to do it in a weakly coupledregime.

IV. WILSON LINES THROUGH THE HORIZON

It was argued above that electric flux measurementsbehave qualitatively the same for a classical and for aquantum wormhole. It is interesting to consider otherprobes involving the gauge field. For example, the classicaleternal black hole also allows a Wilson line to be threadedthrough it. Such Wilson lines have recently been studied ina toy model of holography in [37]. For the black hole,consider extending a Wilson line from the north pole of theleft sphere at r ¼ a through the horizon to the north pole ofthe right sphere:

WER ¼�exp

�iq

ZR

LA

��≈ 1 ð4:1Þ

Since we assume that the gauge field is weakly coupledthroughout the geometry, to leading order we can simplyset it to 0, leading to the approximate equality above.On the other hand, in the entangled spheres case where

there is no geometric connection it is not clear how aWilson line may extend from one box to the other.However, an analogous object with the same quantumnumbers as (4.1) is

WEPR ¼�exp

�iqZ

0

LAL

�ϕLð0Þϕ†

Rð0Þexp�iqZ

R

0

AR

��;

ð4:2Þ

where each Wilson line extends now from the skin of thesphere to the center of the sphere at r ¼ 0, where it ends ona charged scalar field insertion.While the gauge field may be set to zero as in the black

hole case above, we must furthermore account for themixed correlator of the scalar field, which is nonzero onlydue to entanglement. Details of the computation and a plotof the results can be found in Appendix A. The leadinglarge β behavior is

WEPR ∼ exp

�−ω0β

2

�: ð4:3Þ

As the temperature is decreased the expectation value of theWilson line vanishes, consistent with the idea put forthabove that the gauge field living in the wormhole is subjectto strong quantum fluctuations which become stronger,washing out the Wilson line, as the entanglement isdecreased.

V. CONCLUSION

A classical geometry allows an electric field to be passedthrough it. In this paper we have demonstrated that we canmimic this aspect of a geometric connection usingentangled charged matter alone. We also introduced thewormhole susceptibility, a quantitative measure of thestrength of this connection. For quantum wormholes thissusceptibility is suppressed relative to that for classicalwormholes by factors of the dynamical Uð1Þ gaugecoupling, i.e. by powers of ℏ, but, as we argued inSec. III, there is in principle no impediment for thesusceptibilities to be of the same order. The susceptibilityis defined for any state, but for the thermofield state itdirectly measures the electric field produced when apotential difference is applied across the wormhole.We stress that we are not claiming that there is a smooth

geometry in the quantum wormhole; however, there is acrude sense in this setup in which a geometry emerges fromthe presence of entanglement. We showed that by adjustingthe parameters, two boxes of weakly coupled, entangledcharged particles can mimic an Einstein-Rosen bridge intheir response to electric potentials. The structure we havefound in this highly excited state is similar to that of thevacuum of a two-site Uð1Þ lattice gauge theory, whereentanglement between the sites allows the electric fluxbetween them to fluctuate. This is enough to allow for anonzero susceptibility and is somewhat reminiscent ofideas of dimensional deconstruction [38].The equivalence is only at the coarse level of producing

the same electric flux at the boundary of the boxes; moredetailed observations inside the boxes would quickly revealthat charged matter rather than black holes are present.However, the presence of a nonzero wormhole electricsusceptibility already at weak coupling, along with the factthat the two susceptibilities become similar as the couplingis increased, is compatible with the idea that as we go fromweak to strong coupling entangled matter becomes an ERbridge.It is worth noting that the quantum wormholes consid-

ered still satisfy Gauss’s law (3.4) in that every field lineentering one side must exit from the other. This is due to thecorrelated charge structure of the states considered:

jψi ∼XQ

j −QijQi: ð5:1Þ

If we coherently increase the charge of the left sectorrelative to the right, then this would correspond to having adefinite number of charged particles inside the wormhole.Alternatively, we could consider a more generic state,

involving instead an incoherent sum over all charges. Thistype of generic quantum wormhole does not satisfy anyanalog of Gauss’s law. At first glance, this appears non-geometric, in agreement with the intuition that a genericstate should not have a simple geometric interpretation

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[39,40]. On the other hand, we could also simply state thatwe have filled the wormhole with matter that is not in acharge eigenstate, i.e. a superconducting fluid. Thus some“nongeometric” features nevertheless have an interpreta-tion in terms of effective field theory, and a two-sidedanalog of the holographic superconductor [41–43] mightcapture universal aspects of the gauge field response ofsuch a state.We note also that the susceptibility is constructed from

conserved charges, and so it commutes with theHamiltonian. Thus the time-evolved versions of the ther-mofield state (which have been the subject of much recentstudy as examples of more “generic” states [11,22,29,44–46]) do not scramble charges: they all have the samewormhole susceptibility as the original thermofield stateand precisely satisfy Gauss’s law. We also find that thewormhole susceptibility must be conserved if two discon-nected clouds of entangled matter are collapsed to form twoblack holes, which presumably then must have an Einstein-Rosen bridge between them. This provides a crude reali-zation of the collapse experiment proposed in [11].It is of obvious interest to generalize our considerations

to gravitational fields. In that case the wormhole gravita-tional susceptibility corresponding to (2.15) directly mea-sures Newton’s constant in the wormhole throat. Note alsothat the form of the “ER ¼ EPR” correspondence studiedhere requires the existence of perturbative matter chargedunder every low-energy gauge field: e.g. to form a quantumwormhole to have a nonzero wormhole magnetic suscep-tibility and thus to admit magnetic fields, we would requireentangled magnetic monopoles. If the charge spectrumwere not complete, one could certainly tell the differencebetween an ER bridge and an EPR one. Precisely such acompleteness of the charge spectrum in consistent theoriesof quantum gravity has been conjectured on (somewhat)independent grounds [32,33,47].Finally, we find it intriguing that the two computations

performed here result in qualitatively similar answers, butarise from different sources and at different orders in bulkcouplings. One might be tempted to speculate that in aformulation of bulk quantum gravity that is truly non-perturbative, these two very different computations couldbe understood as accessing a more general concept thatreduces in different limits to either perturbative entangle-ment or classical geometry. It remains to be seen what thismore general concept might be.

ACKNOWLEDGMENTS

It is our pleasure to acknowledge helpful discussionswith J. Camps, N. Engelhardt, D. Harlow, D. Hofman, P.Kraus, P. de Lange, L. Kabir, M. Lippert, A. Puhm, J.Santos, G. Sarosi, J. Sully, L. Susskind, D. Tong, L.Thorlacius, and B. Way. This work was supported bythe National Science Foundation Grant No. DGE-0707424,the University of Amsterdam, the Netherlands Organisation

for Scientific Research (NWO), and the D-ITP consortium,a program of the NWO that is funded by the Dutch Ministryof Education, Culture and Science (OCW). N. I. would liketo acknowledge the hospitality of DAMTP at the Universityof Cambridge while this work was in progress.

APPENDIX: CHARGED SCALARFIELD COMPUTATIONS

Here we present some details of the charged scalar fieldcomputations presented in the main text. Similar resultswould be obtained for essentially any system in anygeometry, but for concreteness we present the preciseformulas for the charged scalar field in a spherical box.The relevant part of the action is

Sϕ ¼ −Z

d4xðjDϕj2 þm2jϕj2Þ: ðA1Þ

The scalar field is confined to a spherical box of radius awith Dirichlet boundary conditions ϕðr ¼ aÞ ¼ 0. We firstcompute the single-particle energy levels.Expanding the field in spherical harmonics as ϕ ¼PlmpϕlpðrÞe−iωtYlmðθ;ϕÞ we find the mode equation

for ϕlpðrÞ to be

1

r2∂rðr2∂rϕlpðrÞÞ −

lðlþ 1Þr2

ϕlpðrÞ ¼ ðm2 − ω2ÞϕlpðrÞ:ðA2Þ

Here p is a radial quantum number and l is angularmomentum as usual. The normalizable solutions to theradial wave equation are spherical Bessel functions oforder l:

ϕlpðrÞ ¼ clpjsðl; λlprÞ clp ¼ 2ffiffiffiffiffiffiffiffia3π

p ðJlþ32ðλlpaÞÞ−1:

ðA3Þ

The normalization clp has been picked such that

Xp

ϕlpðrÞϕlpðr0Þ ¼δðr − r0Þ

r2: ðA4Þ

In clp, JνðxÞ is an ordinary Bessel function of the first kind.Imposing the Dirichlet boundary condition fixes λp ¼ xlp

a ,where xlp is the p-th zero of the l-th spherical Besselfunction. This determines the energy levels to be

ωlp ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2 þ

�xlpa

�2

s: ðA5Þ

We are now interested in computing the charge suscep-tibility at finite temperature T and chemical potential μ.

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From elementary statistical mechanics we have the usualexpression for the charge

hQi¼qXlp

ð2lþ1Þ�

1

1−eβðωlpþqμÞ−1

1−eβðωlp−qμÞ

�; ðA6Þ

where we have included the degeneracy factor ð2lþ 1Þ.Linearizing this in μ we obtain (3.5), where it is understoodthat the sum over single-particle states there includes a sumover angular momentum eigenstates:

Pn →

Plpð2lþ 1Þ.

Next we compute the correlation function hϕ†Lð0ÞϕRð0Þi

across the two sides of the thermofield state (3.3) (with

μ → 0). The fastest way to compute this is to note that thetwo sides of the thermofield state can be understood asbeing connected by Euclidean time evolution through β

2.

Thus the mixed correlator can be calculated by computingthe usual Euclidean correlator between two points sepa-rated by β

2in Euclidean time (see e.g. [1]). If the single-

particle energy levels are given by ωpl, then the Euclideancorrelator between two general points is

Gðτ; r; θ;ϕ; τ0; r0; θ0;ϕ0Þ

¼Xlmp

1

2ωlp

cosh ðωlpðτ − τ0 − β2ÞÞ

sinhðβωlp

2Þ

× ϕplðrÞϕplðr0ÞYlmðθ;ϕÞY�lmðθ0;ϕ0Þ; ðA7Þ

where in this expression the normalization of the modefunctions (A4) is important.For our application to the Wilson line in (4.2) we care

about the specific case τ − τ0 ¼ β2and r ¼ r0 ¼ 0. The

spherical Bessel functions with nonzero angular momen-tum l ≠ 0 all vanish at the origin r ¼ 0. Thus the sum isonly over the l ¼ 0 modes. The result of performing thissum numerically is shown in Fig. 6, but it is easy to see thatat small temperatures the answer will be dominated by thelowest energy level and is

hϕLð0Þ†ϕRð0Þi ∼ exp�−ω0β

2

�: ðA8Þ

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0.5 1.0 1.5 2.0 2.5m

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log L R†

FIG. 6. Numerical evaluation of logarithm of hϕLð0Þϕ†Rð0Þi,

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