The Replica Trick and Replica Wormholes

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MSc Physics and Astronomy

Theoretical Physics

Master Thesis

The Replica Trick and Replica Wormholes

by

Joost Pluijmen

6044557

August 2020

60 ECTS

Research carried out between May 2019 and August 2020

Supervisor/Examiner:

Examiner:

Dr. Ben Freivogel

Prof. Dr. Erik Verlinde

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A B S T R A C T

Stephen Hawking proposed that a black hole formed from the collapse of a pure state would disappear after a period of evaporation. The resulting evaporated radiation, known as Hawking radiation, would then be left in a mixed state. In case this is true, information would be lost and there would be no unitary evolution from the initial pure state to a final pure state. This problem is called the ”black hole information paradox” (BHIP). If we demand unitarity, the entropy that is created during the formation and evaporation of a black hole should follow the ”Page curve” as the black hole evaporates. In this thesis, I will give an introduction to the concept of entropy and show examples of how we can use the ”replica trick” to calculate this entropy. The replica trick can also be used to calculate the entropy of evaporating Euclidean black holes with the help of ”replica wormholes”. The idea is that by introducing these wormholes, the Hawking radiation will end up in a pure state so the we can solve the BHIP. Here, we show how these replica wormholes ensure unitarity in AdS2.

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Tingelingeling — Pieter H. M. Pluijmen

A C K N O W L E D G M E N T S

Many thanks to my supervisor Ben Freivogel, who helped me outstandingly with all his advises. Thanks to Ben, the trajectory of my thesis project has been really smooth. The Zoom-talks during the Corona pandemic where good replacements of the earlier face-to-face meetings, with the blackboard from a distance, to help me understand different topics. I also want to thank my girlfriend Esin, who dragged me trough the last period of my study and who supports me in every aspect of life. Last but not least I want to thank my parents who are my oldest and dearest friends and who have had my back during my whole studentship.

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C O N T E N T S

1 i n t r o d u c t i o n 1

2 v o n n e u m a n n e n t r o p y a n d t h e r e p l i c a t r i c k 4

2.1 Von Neumann Entropy . . . 4

2.2 The Replica Trick . . . 6

2.3 Canonical Ensemble . . . 6

2.4 Example: the Harmonic Oscillator . . . 9

3 e u c l i d e a n pat h i n t e g r a l s a n d t h e r e p l i c a t r i c k i n q ua n t u m m e c h a n i c s 11 3.1 Propagator . . . 11 3.2 Euclidean Time . . . 12 3.3 Free Particle . . . 15 3.3.1 Propagator . . . 15 3.3.2 Replica Trick . . . 18 3.4 Harmonic Oscillator . . . 19 3.4.1 Propagator . . . 19 3.4.2 Replica Trick . . . 23 4 e u c l i d e a n pat h i n t e g r a l s a n d t h e r e p l i c a t r i c k i n f r e e s c a l a r f i e l d t h e o r y 28 4.1 Propagator . . . 28 4.2 Euclidean Spacetime . . . 30

4.3 Massive Free Scalar Field in d-Dimensional Euclidean Spacetime . . . 32

4.3.1 Replica Trick . . . 36

4.4 The Cardy Formula . . . 38

4.5 Rindler Coordinates . . . 41

4.5.1 Minkowski spacetime . . . 41

4.5.2 Euclidean spacetime . . . 43

4.6 Massless Free Scalar Field in 2-dimensional Euclidean Rindler Spacetime 44 5 b l a c k h o l e e n t r o p y a n d r e p l i c a w o r m h o l e s 50 5.1 Black Hole Entropy and the Entropy of Hawking Radiation . . . 50

5.1.1 Bekenstein-Hawking entropy . . . 50

5.1.2 Partition function . . . 51

5.1.3 Effective action . . . 51

5.1.4 New Rules . . . 52

5.2 Euclidean AdS₂ . . . 53

5.3 Replica Wormholes in Euclidean AdS₂ . . . 55

5.3.1 Constructing fM_n . . . 55

5.3.2 IdentifyingM_n . . . 62

5.4 Replica Trick on the Replica Manifold . . . 63

6 c o n c l u s i o n s & future research 65 6.1 Conclusions . . . 65

6.2 Future Research . . . 66

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c o n t e n t s v

a.1 Replica Trick . . . 67

a.2 Saddle Point Approximation . . . 68

a.2.1 Quantum Mechanics . . . 68

a.2.2 Free Scalar Field Theory . . . 73

a.3 Direct Calculation of the Propagator of the Free Particle . . . 75

a.4 Direct Calculation of the Partition Function of the Harmonic Oscillator . 76

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1

I N T R O D U C T I O N

Almost 50 years after Stephen Hawking discovered that black holes evaporate in the form of Hawking radiation [1,2], theoretical physicists are still not sure about what happens

to quantum information in and around black holes. This uncertainty is most clearly explained in the Black Hole Information Paradox (BHIP) [3,4,5].

An important feature of the BHIP involves the evolution of the state of the system. This state can be represented by the density matrix ρ, which describes the possible configurations of the system. The Von Neumann entropy SVNdiagnoses the ”mixedness”

of a state and it vanishes when ρ is in a pure state1

. An important property of the Von Neumann entropy is that it is invariant under unitary time evolution. This unitarity implies that a state that is pure at an initial time will remain pure after a unitary time evolution. The BHIP, however, signals that when a black hole, formed from a pure state, evaporates, the Hawking radiation will end up in a mixed state, which means that unitarity is violated. This effect will cause the Von Neumann entropy to keep growing, whereas unitarity implies that the final density matrix should be pure so that the final Von Neumann entropy of the Hawking radiation should decrease.

The Bekenstein-Hawking entropy, SBH = _4GA_N, describes the entropy of a black hole

as a thermodynamic object2

. Here, A represents the area of the horizon of the black hole, whereas GN is the Newton constant. When a black hole evaporates, logically, SBH

decreases. This means that at some point in time, the entropy of the Hawking radiation S_rad, should equal SBH. Following the work of Don Page [6, 7], we call this time the

Page time tPage. Page demonstrated that the entropy should follow the Page curve. In

Figure 1we can see that the green line representing the entropy of the Hawking radiation keeps increasing over time, whereas the orange line representing the Bekenstein-Hawking entropy of the black hole decreases to zero. The purple line represents the Page curve and shows how the entropy of the combination of the black hole and the Hawking radiation is expected to evolve under unitary time evolution.

In a series of papers [9,10,11,12,13, 14,15], a set of ”new rules” was developed to

calculate the entropy of evaporating black holes and of the Hawking radiation. These rules involve a so-called ”quantum extremal surface” that is not the event horizon of the black hole. This surface minimises the generalised entropy Sgen which combines the

Bekenstein-Hawking entropy with the entropy of matter in and around the black hole. An important facet of this new concept is that it seems to be consistent with unitary time evolution.

In this thesis, I will examine the properties of the density matrix in the context of quantum mechanics and of free scalar field theory. As stated before, the density matrix defines the Von Neumann entropy, but in many cases is very had to compute. A method

1

In Chapters 2, 3 & 4, I will use S instead of SVNto indicate the Von Neumann entropy. 2

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i n t r o d u c t i o n 2

Figure 1: Schematic behaviour of the entropy of a black hole and of the Hawking radiation (Figure 7from [8])

to compute the Von Neumann entropy is the ”replica trick”. A useful property of the replica trick is that it mostly uses the partition function Z instead of the density matrix. The partition function can be computed by performing Euclidean path integrals. Here, Euclidean indicates that the time dimension on which the path integral is computed, is Wick rotated. This means that the time direction becomes imaginary. When this imaginary time is periodic, it indicates thermal behaviour.

The replica trick can also be performed to compute the entropy of evaporating black holes. In this case, it is possible to imagine that different copies of a Euclidean back hole are connected with each other through ”replica wormholes”. These replica wormholes can arise in the gravitational path integral by the introduction of dynamical objects that can connect the copies with each other. In this thesis, we explain this with the introduction of cosmic branes. These dynamical objects can appear when the metric is dynamical, which is the case in gravitational regions. They turn out to be consistent with the new rules as explained above, so that when we perform the replica trick on the replica manifold, we can retrieve the right entropy for the Hawking radiation [13,16].

Outline

Inchapter 2the concept of the Von Neumann entropy will be explained in terms of the density matrix. In the canonical ensemble, the Von Neumann entropy can be computed with the help of the partition function which turns out to be a more useful method. A generalised version of the Von Neumann entropy can be be given by the so-called Renyi entropies, which can be used to perform the replica trick. The chapter concludes with an example of how to compute the replica trick for a simple harmonic oscillator.

Inchapter 3, the concept of Euclidean path integrals in quantum mechanics is explained. A useful method to compute these path integrals is the saddle point approximation. In Euclidean time, the path integrals have a close connection to the density matrix and the partition function, so that they can be used to compute the thermodynamic properties of quantum mechanical systems. The chapter concludes with examples of the Euclidean

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i n t r o d u c t i o n 3

path integrals for the free particle and the harmonic oscillator, where for both cases, the replica trick can be performed.

In chapter 4the concept of Euclidean path integrals is explained for free scalar field theories. Similar as with the quantum mechanical path integrals, these path integrals can be used to compute the thermodynamic properties of field theories. The saddle point approximation will be used to compute the path integrals of the massive free scalar field in d-dimesional Minkowski spacetime. A special representation of Minkowski spacetime is given by the Rindler metric of an uniformly accelerated observer. The last example involves the computation of the entropy of a massless free scalar field in 2-dimensional Rindler spacetime. As a bonus, this chapter provides a derivation of he Cardy formula for 2-dimensional CFT’s.

In chapter 5the concept of black hole entropy is explained. The generalised entropy formula seems to be insufficient to compute the right entropy for evaporating black holes, so that two ”new rules” are introduced. The introduction of n replica wormholes as saddle points in the Euclidean path integral can be used to perform the replica trick over a spacetime manifold that we constructed in AdS2.

Inchapter 6I will conclude this thesis by giving some final thoughts on the retrieved results. Furthermore, I will give some suggestions for future research.

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2

V O N N E U M A N N E N T R O P Y A N D T H E R E P L I C A T R I C K

i=0

piln pi. (2.1.7)

When the density matrix represents a mixed state, S(ρ) >0, whereas if it represents a

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2.1 von neumann entropy 5

Unitarity

An important property of the Von Neumann entropy is its invariance under unitary time evolution

S(ρ) =S

UρU−1, (2.1.8)

where U(tf, ti)represents a unitary operator. Here we can see that we can relate the

density matrix at a final time tf to the density matrix at an initial time ti as

ρ(tf) =U(tf, ti)ρ(ti)U−1(tf, ti). (2.1.9)

This principle is called unitarity. Joint Systems

The Hilbert space of a joint system AB can be represented by the tensor product of systems A and B

H_AB = H_A⊗ H_B. (2.1.10)

If we consider a density matrix ρAB and we want to find the state of system A, we need

to trace out the other subsystem

ρA =TrBρAB, (2.1.11)

which is called the reduced density matrix. If ρAB is a product state

ρAB =ρA⊗ρB, (2.1.12)

then

hO_A⊗ O_Bi_ρ_AB− hO_Ai_ρ_AhO_Bi_ρ_B =0. (2.1.13)

This means that correlation functions of operators working on A and B vanish so that A and B are uncorrelated. When ρAB is not a product state, ρA and ρB are mixed states

and A and B are correlated. This means that A and B are entangled. The Von Neumann entropy of joint systems is extensive and subadditive,

ρAB =ρA⊗ρB →S(AB) =S(A) +S(B)

ρAB 6=ρA⊗ρB →S(AB) 6=S(A) +S(B)

(2.1.14) and obeys the Araki-Lieb inequality

S(AB) ≥ |S(A) −S(B)|. (2.1.15)

This implies that when ρAB is a pure state, S(AB) =0 and S(A) =S(B). The so-called

mutual information, detects the amount of correlation or entanglement between the subsystems

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2.2 the replica trick 6

2.2 t h e r e p l i c a t r i c k

Renyi entropies Sn are a one-parameter generalisation of the Von Neumann entropy [17],

defined for n ≥ 0 and n 6=1. We can express these entropies in terms of the density matrix ρ or the eigenvalues pn

Sn = ln Tr ρ n

1−n =

ln∑∞_i=0pin

1−n . (2.2.1)

Renyi entropies are related to the Tsallis entropy [18]

Sn,Tsallis= Tr ρn−1 1−n (2.2.2) through Sn,Tsallis = e(1−n)Sn₋₁ 1−n . (2.2.3)

The replica trick implies that we can use the Renyi entropies Sn to retrieve the Von

Neumann by fitting the result to an analytic function of n, and taking the limit n→1, seeA.2for a full derivation. In terms of the density matrix or its eigenvalues this implies

lim n→1Sn =limn→1 ln Tr ρn 1−n =nlim→1 ln∑∞_i=0pni 1−n (2.2.4)

Alternatively, we can define the Von Neumann entropy in terms of the Renyi entropies by various integral identities [19]. The example

S= Z ∞ 0 dx ∞

∑

n=0 (−x)n (n+1)!(Tr ρ n₋₁₎_, _(2.2.5)

makes use of the identity Z ∞ 0 da a (xe −ax₋_xe−a_{) = −}_{x ln x.} _(2.2.6) 2.3 c a n o n i c a l e n s e m b l e

In statistical mechanics, the canonical ensemble of a system represents the possible states of a system that is coupled to a heat bath [19, 20, 21]. The system is in thermal

equilibrium with the heat bath at a fixed temperature T, which is represented by the parameter β = _k1

BT. From now on we will set kB = 1, so that β =

1

T. In the canonical

ensemble, the density matrix is

ρ= e

−βH

Z (2.3.1)

where H is the Hamiltonian of the system. This thermal density matrix maximises the entropy of the system under the condition that the system has some fixed energy hHi =E. The partition function Z is defined as

Z=Tr e−βH ₌ ∞

∑

i=0

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2.3 canonical ensemble 7

and can also be expressed as an integral

Z= Z dx ∞

∑

i=0 hψi|xi hx|e−βH|ψii = Z dx ∞

∑

i=0 hx|e−βH_| ψii hψi|xi = Z dxhx|e−βH_|_x_i_. (2.3.3)

The eigenvalues of H are Ei, so that if we compare2.3.1with2.1.2, we can see that the

eigenvalues of the density matrix are pi =

e−βEi

Z , (2.3.4)

and the density matrix can be expressed as

ρ= 1 Z ∞

∑

i=0 e−βEi_|_ψ ii hψi|. (2.3.5)

In terms of the energy eigenvalues, the partition function is

Z=

∞

∑

i=0

e−βEi_. _(2.3.6)

An important expectation value in the canonical ensemble is the expectation value of the internal energy, normally represented as E, which is a sum over the eigenvalues of ρ times the eigenvalues of H

E= hHi =Tr ρH=

∞

∑

i=0

piEi. (2.3.7)

We can now use2.3.4and2.3.6to see that

2.3.6, we can see that

S=βE+ln Z = −β∂βln Z+ln Z = (1−β∂β)ln Z = −β2∂β ln Z β . (2.3.10)

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2.3 canonical ensemble 8

At finite temperature, the state of the system minimises the so-called ”free energy” F

F= E−TS. (2.3.11)

If we fill in the value for S in2.3.10, we will find an expression for F in terms of Z

F= E−T(βE+ln Z)

= −ln Z

β ,

(2.3.12)

so that we can also express S in terms of F

S= βE−βF. (2.3.13)

We now can express the partition function in terms of the free energy

Z=e−βF_, _(2.3.14)

which tends to be extremely useful when we want to calculate the partition function of black holes.

Replica Trick

In the canonical ensemble, the Renyi entropies are

Sn= 1 1−nln Tr e−nβH Zn = 1 1−nln ∞

∑

i=0 e−nβEi Zn . (2.3.15)

If we now formulate that

Zn =Tr e−nβH = ∞

∑

i=0

e−nβEi_, _(2.3.16)

we can write the Renyi entropies as Sn= 1 1−nln Zn Zn = ln Zn−n ln Z 1−n . (2.3.17)

The replica trick then implies lim n→1Sn=nlim→1 ln Zn−n ln Z 1−n = lim n→1 ∂n(ln Zn−n ln Z) ∂n(1−n) = lim n→1− ∂n(Zn) Zn +ln Z = lim n→1− ∂n(Zn) Z +ln Z (2.3.18)

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2.4 example: the harmonic oscillator 9

2.4 e x a m p l e: the harmonic oscillator

The Hamiltonian of the quantum harmonic oscillator is [20]

H= p

2

2m +

mω2x2

2 . (2.4.1)

Here, x and p respectively denote the position and the momentum operator of the oscillator, m its mass and ω its angular frequency. With the introduction of ladder operators a=r mω 2 x+ i mωp a† =r mω 2 x− i mωp , (2.4.2)

we can write the Hamiltonian of the harmonic oscillator as H= (a†a+ 1

2)ω, (2.4.3)

with energy eigenvalues

Ei = (i+

1

2)ω. (2.4.4)

In the canonical ensemble, the density matrix of the harmonic oscillator can be expressed in terms of its eigenvalues

ρ= 1 Z ∞

∑

i=0 e−β(i+1₂)ω_| ψii hψi| (2.4.5)

Here, the partition function is

Z= ∞

∑

i=0 e−β(i+1₂)ω ₌_e−βω/2

∑

∞ i=0 (e−βω₎i_. _(2.4.6)

Since βω>0 it follows that e−βω_<_{1 so the resulting sum is a geometric series}

∞

∑

i=0

xi = 1

1−x (2.4.7)

so that the partition function becomes

Z= e −βω/2 1−e−βω = 1 eβω/2₋_e−βω/2 = 1 2csch(βω/2). (2.4.8)

We could directly calculate the entropy of the harmonic oscillator using2.3.10

S=β2∂β ω 2 + ln 1−e−βω β ! =β2 β∂β (1−e−βω₎ β2(1−e−βω) − ln 1−e−βω β2 ! = βωe −βω 1−e−βω −ln 1−e−βω_. (2.4.9)

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2.4 example: the harmonic oscillator 10

Replica Trick

For the replica trick, we want to make use of Zn

Zn =

e−nβω/2 1−e−nβω or

1

2csch(nβω/2). (2.4.10)

With the first term in2.4.10, we can perform the replica trick according to2.3.18

lim n→1Sn=nlim→1− ∂n(Zn) Z +ln Z = lim n→1− ∂n e−nβω/2 1−e−nβω e−βω/2 1−e−βω +ln e −βω/2 1−e−βω = βω 2 1+e−βω 1−e−βω − βω 2 −ln 1−e−βω = βωe −βω 1−e−βω −ln 1−e−βω =S (2.4.11)

and retrieve the same entropy as in2.4.9. We can also use the second term in2.4.10and

do the replica trick lim n→1Sn= limn→1− ∂n(Zn) Z +ln(Z) = lim n→1− ∂n 1₂csch(nβω/2) 1 2csch(βω/2) +ln 1 2csch(βω/2) = lim n→1 βω 2 1 2coth(nβω/2)csch(nβω/2) 1 2csch(βω/2) +ln 1 2csch(βω/2) = βω 2 coth(βω/2) +ln 1 2csch(βω/2) = S. (2.4.12)

We can see that this result matches2.4.9if we use the identities 1

2csch(x) = e −_x 1−e−2x and 1 2coth(x) = e −2x 1−e−2x + 1₂.

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3

E U C L I D E A N PAT H I N T E G R A L S A N D T H E R E P L I C A T R I C K I N Q U A N T U M M E C H A N I C S

In this chapter. The calculations are done with the help of [22, 23, 24, 25, 26]. The

provided calculations are based on examples in [24,22,23,25,26,27,28].

3.1 p r o pa g at o r

In the path integral formulation of quantum mechanics, the propagator gives the tran-sition amplitude between the spacetime time points(xi, ti) and (xf, tf)[24, 29]. With

¯h=1, the propagator is

K(x_f, tf; xi, ti) =xf, tf

e−i(tf−ti)H|x_i, t_ii. (3.1.1) Here, the Hamiltonian H is the sum of a kinetic energy term T = _2mp2 and a variable potential energy term V(x)

H = p

2

2m +V(x). (3.1.2)

If we then want to write the transition amplitude in the form of a path integral, we need to ”slice” the time between ti and tf in N equal pieces (for a whole derivation, see for

example [29]). The propagator then becomes the sum over all possible trajectories from

x(ti)to x(tf). In the limit N→∞, the propagator can be written as a path integral

K(x_f, tf; xi, ti) = N

Z x(t_f)=x_f

x(ti)=xi

D[x]eiS[x(t)], (3.1.3)

with normalisation constantN. Here, D[x]represents the integration measure over all possible paths, whereas S[x(t)]is the action

S[x(t)] =

Z t_f

ti

L(x, ˙x)dt. (3.1.4)

The Lagrangian L(x, ˙x) depends on the position operator x and its time derivative

dx

dt = ˙x. We can retrieve the Lagrangian from the Hamiltonian by using the Legendre

transformation

H= ˙x∂L

∂ ˙x −L(x, ˙x) (3.1.5)

and the definition of the momentum

p = ∂L

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3.2 euclidean time 12

When we combine these two definitions, we get H= ˙xp−L(x, ˙x)

→ dH

dp = ˙x

(3.1.7)

From3.1.2, we know that

dH dp = p m, (3.1.8) so that p m = ˙x. (3.1.9)

We can consequently fill in these values in3.1.7, where

L(x, ˙x) = ˙x∂L ∂ ˙x −H (3.1.10) and retrieve L(x, ˙x) = m˙x 2 2 −V(x). (3.1.11)

Now the propagator can be written as K(x_f, t_f; xi, ti) = N Z x(t_f)=x_f x(ti)=xi D[x]ei Rt f ti m˙x2 2 −V(x) dt . (3.1.12) 3.2 e u c l i d e a n t i m e

In the previous sector we saw the regular path integral formulation of quantum mechanics. If we however take a closer look at the propagator, we can see that there is close resemblance to the definition of the density matrix in the canonical ensemble2.3.1. If we

perform a so-called Wick rotation [29] of the time component in the propagator t→ −iτ,

the propagator transforms to

K(xf, τf; xi, τi) =xf, τf

e−(τf−τi)H|x_i, τ_ii. (3.2.1) In the path integral formulation, the exponential term transforms as

exp i Z t_f ti m ˙x2 2 −V(x) dt →exp ( i Z τf τi m 2 dx d(−iτ) 2 −V(x) ! d(−iτ) ) =exp ( − Z τf τi m 2 dx dτ 2 +V(x) ! dτ ) =exp{−IE[x(τ)]}, (3.2.2)

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where we defined the Euclidean action IE[x(τ)]

IE[x(τ)] = Z τf τi m 2 dx dτ 2 +V(x) ! dτ = Z τf τi LE(x, ˙x)dτ, (3.2.3)

with the Euclidean Lagrangian now dependent on ˙x= dx_dτ

LE x,dx dτ = m 2 dx dτ 2 +V(x) ! . (3.2.4)

The propagator can then be written as K(x_f, τf; xi, τi) = N

Z x(_τ_f)=x_f

x(τi)=xi

D[x]e−IE[x(τ)]_. _(3.2.5)

The partition function can be retrieved by tracing over x, so that we can say that we connect the end points of a line, xiand xf. When we let the Euclidean time run from 0 to

nβ Zn = Z dxhx|e−nβH|xi = Z dxK(x, nβ; x, 0) = N Z dx Z x(nβ)=x x(0)=x D[x]e−IE[x(τ)] = N Z x(nβ)=x(0) D[x]e−IE[x(τ)]_. (3.2.6)

When τ runs from 0 to β, we can recognize the density matrix up to a factor of Z in its position representation x_f ρ|x_ii = 1 Zxf e−βH|x_ii = 1 ZK(xf, β; xi, 0) = N Z Z x(β)=xf x(0)=xi D[x]e−IE[x(τ)]_. (3.2.7)

Accordingly, the density matrix now becomes independent of the normalisation constant N xf ρ|x_ii = NRx(β)=xf x(0)=xi D[x]e −IE[x(τ)] NRx(β)=x(0) D[x]e−IE[x(τ)] = Rx(β)=xf x(0)=xi D[x]e −IE[x(τ)] Rx(β)=x(0) D[x]e−IE[x(τ)] . (3.2.8)

The most common way to perform the replica trick is by using the result of Zn and

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Tr ρn. To do so, we first want to compute ρn. If we state that xf =xnand xi = x0, we can

write xf ρn|x_ii = hx_n|ρn|x₀i = Z n−1

∏

i=1 dxihxn|ρ|xn−1i · · · hx1|ρ|x0i. (3.2.9)

We can then compute Tr ρnso that xn= x0

Tr ρn=hx0|ρn|x0i = Z dx0 Z n−1

∏

i=1 dxihx0|ρ|xn−1i · · · hx1|ρ|x0i. (3.2.10)

In most cases, this calculation is much harder than the computation of Zn, but an exact

example is given in3.4.2.

Saddle Point Approximation

A useful method to calculate the path integral, is the saddle point approximation. The precise derivation is displayed in A.2, but the idea is that we can expand the action around a classical path

x(τ) =xcl(τ) +δx(τ) (3.2.11)

where δx(τ) counts for the quantum fluctuations of the classical path. Furthermore,

x_cl(τ_i) =x_i and x_cl(τ_f) =x_f, so that δx(τ_i) =δx(τ_f) =0. The propagator then becomes

K(xf, τf; xi, τi) = Ne−IE[xcl]

Z δx(τf)=0

δx(τi)=0

D[δx]e−IE[δx]. (3.2.12)

When the classical action IE[xcl]dominates the path integral, we can approximate the

propagator as follows

K(xf, τf; xi, τi) ≈e−IE[xcl]. (3.2.13)

If we implement3.2.7and3.2.6, we can write the density matrix as

x_f ρ|x_ii = K(xf, β; xi, 0) R dxK(x, β; x, 0) = Ne−IE[xcl]Rδx(β)=0 δx(0)=0 D[δx]e −IE[δx] NRxi=xf=x_dxe−IE[xcl]Rδx(β)=0 δx(0)=0 D[δx]e −IE[δx] = e −IE[xcl] Rxi=xf=x_dxe−IE[xcl], (3.2.14)

where we can see that the density matrix is independent of the normalisation constant N. Here, the partition function is

Z= N Z x_i=x_f=x dxe−IE[xcl] Z δx(β)=0 δx(0)=0 D[δx]e−IE[δx]. (3.2.15)

If we look at3.2.13, we can see that

Z≈

Z x_i=x_f=x

dxe−IE[xcl]_, _(3.2.16)

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3.3 free particle 15

3.3 f r e e pa r t i c l e

The simplest quantum mechanical path integral is that of the ”free particle”. Here, we will perform this path integral directly in Euclidean time. The Hamiltonian of the free particle is

H= p

2

2m. (3.3.1)

After a Wick rotation and with3.1.9, we can see that the Euclidean action of the free

particle is IE[x(τ)] = m 2 Z τf τi ˙x(τ)2dτ. (3.3.2) 3.3.1 Propagator

A direct derivation of the propagator can be achieved for the free particle and is illustrated inA.3. Here, however, we use the saddle point approximation to retrieve the propagator which gives us the following expression for the action

IE[x(τ)] = m 2 Z τf τi ˙xcl(τ)2dτ+m 2 Z τf τi δ ˙x(τ)2dτ, (3.3.3)

so that the propagator becomes K(xf, τf; xi, τi) = Nexp −m 2 Z τf τi ˙xcl(τ)2dτ Z δx(τf)=0 δx(τi)=0 D[δx]exp −m 2 Z τf τi δ ˙x(τ)2dτ . (3.3.4) We can find an expression for x_cl(τ)by solving the Euler-Lagrange equation for the first

variation of the action for the boundary conditions xcl(τi) =xi and xcl(τf) =xf

∂L(x, ˙x) ∂x(τ) x=xcl − d dτ ∂L(x, ˙x) ∂ ˙x(τ) x=xcl =0 → − d dτm˙xcl(τ) =0 → ¨x_cl(τ) =0, (3.3.5) so that x_cl(τ) = Aτ+B →x_cl(τ) =xi+ xf −xi τf −τi (τ−τi) → ˙xcl(τ) = xf −xi τf −τi . (3.3.6)

Now the classical action can be written as IE[xcl] = m 2 Z τf τi _x f −xi τ_f −τi 2 dτ = m 2 xf −xi 2 τf −τi . (3.3.7)

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Due to the boundary conditions δx(τi) =δx(τf) =0,

−IE[δx] = −m 2 Z τf τi δ ˙x(τ)2dτ = −m 2 δ ˙x(τ)δx(τ)| τf τi + m 2 Z _τ_f τi dτδx(τ) d dτ 2 δx(τ) = m 2 Z _τ_f τi dτδx(τ) d dτ 2 δx(τ) = m 2 Z τf τi dτδx(τ)Oδxˆ (τ), (3.3.8)

where ˆO= _dτd2. Now we can use a simple Gaussian integral to see that

N Z δx(τf)=0 δx(τi)=0 D[δx]e−IE[δx] = N Z δx(τf)=0 δx(τi)=0 D[δx]exp m 2 Z τf τi dτδx(τ)Oδxˆ (τ) = N detn− m 2πOˆ o−1₂ . (3.3.9) To find det−m

2πO we can make a change of variablesˆ

δx(τ) =

∞

∑

n=1

anxn(τ), (3.3.10)

where xn(τ)are the orthonormalized eigenmodes of ˆO so that we can set

ˆ

Oxn(τ) =bnxn(τ). (3.3.11)

The boundary conditions now imply that xn(τ_i) =xn(τ_f) =0. If we change the variables

so that τ0 =τ−τi where τ0 ∈ (0, β)then τi0 =0 and β=τ0f −τi0 then xn(τ0)is a solution

of d dτ0 2 xn(τ0) =bnxn(τ0) →xn(τ0) =A sinh p bnτ0 +B coshpbnτ0 xn(τ_i0 =0) =0 →xn(τ0) =A sinh p bnτ0 xn(τ0_f =β) =0= A sinh p bnβ →sinhpbnβ =0 →pbnβ=iπn→bn =  iπn β 2 →xn(τ0) =A sinh iπn β τ 0 . (3.3.12)

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3.3 free particle 17 To find A, we use Z τ_f0=β τ_i0=0 xn(τ0)xm(τ0)dτ0 =1 → A2 Z _τ0 f=β τ_i0=0 sinh iπn β τ 0 sinh iπm β τ 0 dτ0 =1 → βA 2 2 δnm =1 → A= s 2 β. (3.3.13)

Combining3.3.12and3.3.13and changing back the variables then gives us

n=1 πn τf −τi 2!− 1 2 exp ( −m 2 x_f −xi 2 τf −τi ) . (3.3.18)

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The direct calculation of the propagator as shown inA.3

K(xf, τf; xi, τi) = s m 2π(τf −τi) e− m 2 (_{x f}−_xi)2 τf−_τi . (3.3.19)

can eventually give us the write value for the normalisation constantN

N − m 2π ∞

∏

n=1 πn τ_f −τ_i 2!− 1 2 = s m 2π(τ_f −τ_i) → N = s m 2π(τ_f −τ_i) v u u t− m 2π ∞

∏

n=1 πn τ_f −τ_i 2 . (3.3.20) 3.3.2 Replica Trick

When we let the Euclidean time run from 0 to nβ, we can easily compute the partition function Zn by tracing over the propagator

Zn= Z x_i=x_f=x dxK(x, nβ; x, 0) = r m 2πnβ Z dxe−m2nβ02 = s mL2 2πnβ, (3.3.21)

where L is the length of the system. The density matrix is then xf ρ|x_ii = 1 Z1 K(xf, β; xi, 0) = 1 Le −m(_{x f}−_xi)2 2β _. (3.3.22)

A simple computation then shows that Tr ρ = 1 as expected. For the replica trick, we now want to use2.3.18

lim n→1Sn=nlim→1− ∂n(Zn) Z1 +ln Z1 = lim n→1− ∂n q mL2 2πnβ q mL2 2πβ +ln s mL2 2πβ = lim n→1 1 2n + 1 2ln mL2 2πβ = 1 2 1+lnmL 2 2πβ (3.3.23)

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3.4 harmonic oscillator 19

3.4 h a r m o n i c o s c i l l at o r

The second example of a Euclidean path integral in quantum mechanics, will be the harmonic oscillator. We already know the Hamiltonian of the harmonic oscillator from

2.4.1. After a Wick rotation and with3.1.9, we can see that the Euclidean action of the

harmonic oscillator is IE[x(τ)] = m 2 Z _τ_f τi ˙x(τ)2+ω2x(τ)2 dτ. (3.4.1) 3.4.1 Propagator

We use the saddle point approximation (seeA.2) to retrieve the propagator and find the following Euclidean action

IE[x(τ)] = m 2 Z _τ_f τi ˙x_cl(τ)2+ω2xcl(τ)2 dτ+m 2 Z _τ_f τi δ ˙x(τ)2+ω2δx(τ)2 dτ. (3.4.2)

We can then implement this term in the propagator, so that K(x_f, τf; xi, τi) = Nexp −m 2 Z _τ_f τi ˙x_cl(τ)2+ω2x_cl(τ)2 dτ × Z δx(τf)=0 δx(τi)=0 D[δx]exp −m 2 Z τf τi δ ˙x(τ)2+ω2δx(τ)2 dτ . (3.4.3) The first variation of the action gives the Euler-Lagrange equation of motion

Z _τ_f τi dτ1 δIE[x] δx(τ1) x=xcl δx(τ₁) =0 → ∂L(x, ˙x) ∂x(τ) x=xcl − d dτ ∂L(x, ˙x) ∂ ˙x(τ) x=xcl =0 →mω2xcl(τ) − d dτmx˙cl(τ) =0 →ω2xcl(τ) =x¨cl(τ). (3.4.4)

Now we can integrate the Euclidean classical action by parts and implement the previous results, to get IE[xcl] = m 2 xcl(τ)˙xcl(τ)| τf τi − m 2 Z τf τi xcl(τ)¨xcl(τ) −ω2xcl(τ)2 dτ = m 2 xcl(τ)˙xcl(τ)| τf τi . (3.4.5)

We can find the solution for xcl(τ) by solving the Euler-Lagrange equation for the

boundary conditions x_cl(τi) =xi and xcl(τf) =xf

ω2xcl(τ) = x¨cl(τ) →xcl(τ) = A sinh(ωτ) +B cosh(ωτ) →xcl(τ) = xisinh ω(τf −τ)−xf sinh(ω(τi−τ)) sinh ω(τf −τi) → ˙xcl(τ) = −a xicosh ω(τf −τ)−xf cosh(ω(τi−τ)) sinh ω(τf −τi) , (3.4.6)

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so that the Euclidean classical action can be written as IE[xcl] = m 2 xcl(τ)˙xcl(τ)| τf τi = mω 2 sinh ω(τf −τi) (x2_f +x2_i)cosh ω(τf −τi) −2xixf . (3.4.7)

The second term in the action can be retrieved by again an integration by parts and with the boundary conditions δx(τi) =δx(τf) =0

−IE[δx] = −m 2 Z _τ_f τi dτ δ ˙x(τ)2+ω2δx(τ)2 = −m 2 δ ˙x(τ)δx(τ)| τf τi + m 2 Z τf τi dτδx(τ) d dτ 2 −ω2 ! δx(τ) = m 2 Z τf τi dτδx(τ) d dτ 2 −ω2 ! δx(τ) = m 2 Z _τ_f τi dτδx(τ)Oˆωδx(τ). (3.4.8) Here, ˆOω= d dτ 2 −ω2 is defined, so that N Z δx(τf)=0 δx(τi)=0 D[δx]e−IE[δx] = N Z δx(τf)=0 δx(τi)=0 D[δx]exp m 2 Z τf τi dτδx(τ)Oˆωδx(τ) = N detn− m 2πOˆω o−1₂ . (3.4.9)

Just like we did with the example of the free particle, we can express δx(τ)as a sum over

eigenvalues an, so that δx(τ) = ∞

∑

n=1 anxn(τ), (3.4.10)

where xn(τ)are the orthonormalized eigenmodes of ˆOω. We can then consequently set

ˆ

Oωxn(τ) =bnxn(τ). (3.4.11)

According to the boundary conditions, xn(τ_i) = xn(τ_f) = 0. If we make a change of

variables where τ0 = τ−τi with τ0 ∈ (0, β)so that τi0 = 0 and where β=τf0 −τi0, then

xn(τ0)is a solution of d dτ0 2 −ω2 ! xn(τ0) =bnxn(τ0) → xn(τ0) =A sinh p bn+ω2τ0 +B coshpbn+ω2τ0 xn(τ_i0 =0) =0 → xn(τ0) =A sinh p bn+ω2τ0 xn(τ0_f =β) =0= A sinh p bn+ω2β →pbn+ω2β= iπn →bn= iπn β 2 −ω2= − πn β 2 +ω2 ! → xn(τ0) =A sinh iπn β τ 0 . (3.4.12)

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To find the correct value for A, we want to normalize the eigenmodes Z τ_f0=β τ_i0=0 xnxmdτ0 =1 → A2 Z _τ0 f=β τ_i0=0 sinh  iπn β τ 0 sinh iπm β τ 0 dτ0 =1 → βA 2 2 δnm =1 → A= s 2 β, (3.4.13) so that xn(τ0) = s 2 βsinh  iπn β τ 0 . (3.4.14)

We can now fill in the values for β and τ0 to retrieve the right expression for xn(τ)

xn(τ) = s 2 τ_f −τi sinh iπn τ_f −τi (τ−τi) . (3.4.15)

n=1 πn τ_f −τi 2 sinh (τ_f −τi)ω (τ_f −τi)ω . (3.4.20)

We can now combine the results in 3.4.7 and 3.4.20 with 3.4.3 to find the following

expression for the propagator

K(x_f, τf; xi, τi) = N − m 2π ∞

∏

n=1 πn τf −τi 2 sinh (τf −τi)ω (τf −τi)ω !−1₂ ×exp ( − mω 2 sinh ω(τ_f −τ_i) (x2_f +x2_i)cosh ω(τ_f −τ_i)−2x_ix_f ) . (3.4.21) We know from3.3.20the value for the normalisation constant

N = s m 2π(τ_f −τ_i) v u u t− m 2π ∞

∏

n=1 πn τ_f −τ_i 2 , (3.4.22)

so that we finally retrieve the exact propagator of the harmonic oscillator

K(xf, τf; xi, τi) = s m 2π(τf −τi) v u u u u t −m 2π ∏∞n=1 πn τf−τi 2 −m 2π∏∞n=1 πn τf−τi 2 sinh₍₍_τ f−τi)ω) (τf−τi)ω ×exp ( − mω 2 sinh ω(τ_f −τi) (x2_f +x2_i)cosh ω(τ_f −τi) −2xixf ) = s mω 2π sinh (τf −τi)ω ×exp ( − mω 2 sinh ω(τf −τi) (x2_f +x2_i)cosh ω(τf −τi) −2xixf ) . (3.4.23)

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3.4.2 Replica Trick

In order to perform the replica trick, we must find the partition function or the density matrix. One way to retrieve the partition function is by performing a direct calculation as is shown inA.4. Here, however, we focus on retrieving it from the propagator in3.4.23.

We will give two examples of how we can perform the replica trick. The first example involves the use of Zn, while the second example shows how we can perform Tr ρn.

Replica Trick with the use of Zn

We can calculate Znwith the propagator in3.4.23. We let the Euclidean time run from 0

to nβ and we trace over the propagator so that xi =xf =x

Zn = Z x_i=x_f=x dxK(x, nβ; x, 0) = r mω 2π sinh(nβω) Z dx exp − mω 2 sinh(nβω) 2x 2_cosh₍_nβω_{) −}_2x2 = r _mω 2π sinh(nβω) Z dx exp −mωx2cosh(nβω) −1 sinh(nβω) = r mω 2π sinh(nβω) s πsinh(nβω) mω(cosh(nβω) −1) = s 1 2(cosh(nβω) −1). (3.4.24)

Using the identity cosh(2x) −1=2 sinh2(x)gives

Zn= s 1 4 sinh2(nβω/2) = 1 2 sinh(nβω/2) = 1 2csch(nβω/2), (3.4.25)

which is the same result as in2.4.10. We already performed the replica trick with Z_nin 2.4.12.

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Replica Trick with the use of Tr ρn

The density matrix can be computed by using the regular definition xf ρ|x_ii = 1 ZK(xf, τf; xi, τi) =2 sinh(βω/2) r mω 2π sinh(βω) ×exp − mω 2 sinh(βω) (x2_f +x2_i)cosh(βω) −2xixf = s 2mω sinh2(βω/2) πsinh(βω) exp − mω 2 sinh(βω) (x2_f +x2_i)cosh(βω) −2xixf = s mω(cosh(βω) −1) πsinh(βω) exp − mω 2 sinh(βω) (x2_f +x2_i)cosh(βω) −2xixf = r mω tanh(βω/2) π exp − mω 2 sinh(βω) (x2_f +x2_i)cosh(βω) −2xixf , (3.4.26) where we used the identity cosh_sinh(x₍)−_x₎1 =tanh x₂. As we showed in3.2.14, we could have

directly calculated the density matrix without knowing the normalisation constant of the propagator x_f ρ|x_ii = e−IE[xcl] Rxi=xf=x_dxe−IE[xcl] = exp n −_{2 sinh}mω₍ βω) (x2_f +x2_i)cosh(βω) −2x_ix_f o R dx expn−_{2 sinh}mω₍ βω)(2x 2_cosh₍_βω_{) −}_2x2₎o = expn− mω 2 sinh(βω) (x2_f +x2_i)cosh(βω) −2xixf o R dx expn−mωx2_tanhβω 2 o = r mω tanh(βω/2) π exp − mω 2 sinh(βω) (x2_f +x2_i)cosh(βω) −2x_ix_f , (3.4.27) where we again used the identity cosh_sinh(x₍)−_x₎1 =tanh x₂. To perform the replica trick, we first want to compute ρn_{with the help of}3.2.9and fill in the values of density matrix, so

that hxn|ρn|x0i = Nn Z n−1

∏

i=1 dxiexp ( −mω x 2 n+x2n−1 cosh(βω) −2xn−1xn 2 sinh(βω) ) × · · · ×exp ( −mω x 2 1+x20 cosh(βω) −2x0x1 2 sinh(βω) ) , (3.4.28) where we defined N = r mω tanh(βω/2) π . (3.4.29)

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We can simplify this result by summing over all terms in the exponential hxn|ρn|x0i = Nn Z n−1

∏

i=1 dxi ×exp    −mω x2_n+2∑_in₌−₁1x2_i +x2₀cosh(βω) −2∑_nn−₌1₀x_ix_i+1 2 sinh(βω)    . (3.4.30) Now we want to perform Tr ρnwith3.2.10so that

If we then fill in the right value forN, we retrieve hx0|ρn|x0i =  mω tanh(βω/2) π n₂  2π sinh(βω) mω n₂ s 1 det uij =4 sinh2(βω/2) n₂ s 1 det uij =2nsinhn(βω/2) s 1 det uij . (3.4.36)

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For convenience, we will call the determinant of uij, Dnand express it as a n×n matrix

Dn =det uij=           2 cosh(βω) −1 0 · · · 0 −1 −₁ _{2 cosh}(βω) −1 · · · 0 0 0 −1 2 cosh(βω) · · · 0 0 .. . ... ... . .. ... ... 0 0 0 · · · 2 cosh(βω) −1 −1 0 0 · · · −1 2 cosh(βω).           (3.4.37) We can then find the following recursion relation

Dn =2(cosh(βω)Cn−1−Cn−2−1) (3.4.38)

where Cnis the determinant of a submatrix

Cn=        2 cosh(βω) −1 0 · · · 0 −1 2 cosh(βω) −1 · · · 0 0 −1 2 cosh(βω) · · · 0 .. . ... ... . .. ... 0 0 0 · · · 2 cosh(βω).        (3.4.39)

The recursion relation for Cnis

Cn=2 cosh(βω)Cn−1−Cn−2, (3.4.40)

so that we can simplify the recursion relation for Dn

Dn =Cn−Cn−2−2. (3.4.41)

If we want to solve the recursion relation for Cn, we have to solve

x2−2 cosh(βω)x+1=0 → x=cosh(βω) ±sinh(βω) =e±βω_, (3.4.42) so that Cn= Aeβωn+Be−βωn. (3.4.43)

Now we can use the values for C0and C1to solve for Cn

C0=1= A+B→A=1−B →Cn= (1−B)eβωn+Be−βωn =eβωn₋_B₍_eβωn₋_e−βωn₎ C1=2 cosh(βω) =eβω+e−βω =eβω−B(eβω−e−βω) →B= − e−βω eβω₋_e−βω →Cn=eβωn+ e−βω eβω₋_e−βω(e βωn₋_e−βωn₎ =eβωn₊e βω(n−1)₋_e−βω(n+1) eβω₋_e−βω = e βω(n+1)₋_e−βω(n+1) eβω₋_e−βω . (3.4.44)

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By plugging these values into the recursion relation for Dn, we retrieve

Dn= eβω(n+1)₋_e−βω(n+1) eβω₋_e−βω − eβω(n−1)₋_e−βω(n−1) eβω₋_e−βω −2 = e βω(n+1)₋_e−βω(n+1)₋_eβω(n−1)₊_e−βω(n−1) eβω₋_e−βω −2 = e βω₍_eβωn₊_e−βωn_{) −}_e−βω₍_eβωn₊_e−βωn₎ eβω₋_e−βω −2 =eβωn₊_e−βωn₋₂ =2 cosh(βωn) −2 =4 sinh2(βωn/2) (3.4.45)

Since Dn=det uij, we can fill in its value in3.4.36

Tr ρn=hx0|ρn|x0i =2nsinhn(βω/2) s 1 det uij =2nsinhn(βω/2) s 1 4 sinh2(nβω/2) = 2 n_sinhn₍ βω/2) 2 sinh(nβω/2). (3.4.46)

Here we can recognise

Tr ρn= Zn

Zn (3.4.47)

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4

E U C L I D E A N PAT H I N T E G R A L S A N D T H E R E P L I C A T R I C K I N F R E E S C A L A R F I E L D T H E O R Y

This chapter describes the formulation of path integrals in non-interacting or ”free” scalar field theory, which is a branch of quantum field theory. Here, ”free” means that the scalar fields in the theory don’t interact with each other. In the previous chapter, we gave an inside in the path integral formulation in quantum mechanics, which we will generalise to a free scalar field theory here. The calculations in this chapter are based on [17,19,21,30,31,32,33,34,35]

4.1 p r o pa g at o r

In quantum field theory, the geometry of the background spacetime plays a major role. In regular quantum field theory, most calculations are done on a 4-dimensional Minkowski spacetime. However, when we want to formulate a field theory on a curved spacetime as for example in the proximity of a black hole, the calculations become rather different. The line element of a spacetime can be expressed as

ds2 =gµν(x)dxµdxν. (4.1.1)

Here, ds represents the distance between two points in the spacetime. Furthermore, x= xµ = (x0,· · · , xd−1)and gµν(x)is the metric, satisfying

gµσgσν =δµν. (4.1.2)

The standard action for a free scalar field φ(x)can be expressed as I[φ(x)] = Z ddxL φ(x),∇µφ(x), g µν₍_x₎ = Z ddx q |g(x)|L₀ φ(x),∇µφ(x), gµν(x) . (4.1.3) Here,L

φ(x),∇µφ(x), gµν(x) represents the Lagrangian density

L φ(x),∇µφ(x), gµν(x) = q |g(x)| −1 2g µν₍_x_)∇ µφ(x)∇νφ(x) −V[φ(x)] , (4.1.4) where g(x) = det gµν(x)

and V[φ(x)] is the potential. Here, ∇µ is the covariant

derivative, which in case of a scalar field is just the partial derivative

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4.1 propagator 29

The partial derivative is defined as ∂µ = _∂x∂_µ =

∂ ∂x0,· · · , ∂ ∂xd−1

. We can then write the action as I[φ(x)] = Z ddx q |g(x)| −1 2g µν₍_x₎ ∂µφ(x)∂νφ(x) −V[φ(x)] . (4.1.6) Minkowski Spacetime

As stated before, Minkowski spacetime or flat spacetime is the most common background of a quantum field theory. The metric of a d-dimensional Minkowski spacetime can be expressed as a d×d diagonal matrix

gµν(x) =gµν(x) =      −1 0 · · · 0 0 1 · · · 0 .. . ... . .. ... 0 0 · · · 1.      (4.1.7)

In Minkowski spacetime, the metric is mostly known as gµν(x) = ηµν(x). The line

element is gµν(x)dxµdxν = −dx20+ d−1

∑

n=1 dx2_n, (4.1.8)

where x0=t is the time coordinate and xn are the Euclidean space coordinates. Now, we

can write x= xµ = (t, x1,· · · , xd−1) = (t,~x)where~x= (x1,· · · , xn)with 1≤n≤ d−1,

so that

gµν(x)dxµdxν= −dt2+d~x2. (4.1.9)

We can express the propagator in non-interacting scalar field theory in d-dimensional Minkowski space as a quantum mechanical path integral from a field at an initial time

φ(ti,~x) =φi(~x)to a field at a final time φ(tf,~x) =φf(~x)

φ(t_f,~x)e−i(tf−ti)H|φ(t_i,~x)i = N

Z _φ(t_f,~x)=_φ_f(~x)

φ(ti,~x)=φi(~x)

D[φ(x)]eiI[φ(x)]. (4.1.10)

The action in Minkowski spacetime is I[φ(x)] = Z t_f ti dtL(t), (4.1.11) where L(t) = Z ∞ −∞dx1· · · Z ∞ −∞dxnL [φ(x), ∂φ(x)] = Z ∞ −∞d~xL [φ(x), ∂φ(x)], (4.1.12) so that I[φ(x)] = Z t_f ti dt Z ∞ −∞d~xL [φ(x), ∂φ(x)] = Z t_f ti dt Z ∞ −∞d~x q |g(x)| −1 2g µν₍_x₎ ∂µφ(x)∂νφ(x) −V[φ(x)] . (4.1.13)

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4.2 euclidean spacetime 30 Here|g(x)| =1 and ∂µ = ∂ ∂t, ∂ ∂x1,· · · , ∂ ∂xn =∂ ∂t, ∂ ∂~x , so that I[φ(x)] = Z t_f ti dt Z ∞ −∞d~x " 1 2 ∂φ(x) ∂t 2 − 1 2 ∂φ(x) ∂~x 2 −V[φ(x)] # . (4.1.14)

Saddle Point Approximation

Like we did in the quantum mechanical case, we will make use of the saddle point approx-imation. We refer toA.2for a complete derivation. In the saddle point approximation, the action can be written as

I[φ(x)] = I[φcl(x)] + 1 2 Z ddxδφ(x) −V00[φcl(x)] δφ(x) + O(δφ3), (4.1.15)

where the Alembertian gµν

∂µ∂ν = ∂µ∂µ = = −∂ 2 ∂t2 +∑ d n=1 ∂ 2 ∂x2n = −∂ 2 t +∂~2x. The

Euler-Lagrange equation gives 1 p |g(x)|∂µ q |g(x)|gµν ∂νφcl(x) −V0[φ_cl(x)] = φ_cl(x) −V0[φ_cl(x)] = 0. (4.1.16)

Because of the variation of the actionD[φ] → D[δφ], so that the propagator becomes

φ(t_f,~x)e−i(tf−ti)H|φ(t_i,~x)i = N Z _φ(t_f,~x)=_φ_f(~x) φ(ti,~x)=φi(~x) D[φ(x)]eiI[φ(x)] = N Z δφ(x)=0∀x∈∂Ω D[δφ(x)]ei(I[φcl(x)]+ 1 2 R dd_xδφ₍_x₎₍₋_V00_[ φcl(x)])δφ(x)+O(δφ3)) ' NeiI[φcl(x)] Z δφ(x)=0∀x∈∂Ω D[δφ(x)]e i 2 R dd_xδφ₍_x₎₍₋_V00_[_φ cl(x)])δφ(x)_. (4.1.17) 4.2 e u c l i d e a n s pa c e t i m e

Just like we did to the time direction in quantum mechanics, we can perform a Wick rotation on the Minkowski time direction where t→ −iτ so that the metric becomes

gµν(x)dx

µ_dxν_{= −}_d₍₋_iτ₎2₊_d_~_x2 ₌_dτ2₊_d_~_x2_. _(4.2.1)

We then have a d-dimensional Euclidean spacetime which can again be expressed as a d×d diagonal matrix gµν(x) =gµν(x) =      1 0 · · · 0 0 1 · · · 0 .. . ... . .. ... 0 0 · · · 1.      (4.2.2)

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4.2 euclidean spacetime 31

The Wick rotation of the action gives I[φ(x)] t=−iτ = Z it_f iti d(−iτ) Z ∞ −∞d~x " 1 2 ∂φ(−iτ,~x) ∂(−iτ) 2 −1 2 ∂φ(−iτ,~x) ∂~x 2 −V[φ(−iτ,~x)] # = −i Z τf τi dτ Z ∞ −∞d~x " −1 2 ∂φ(x) ∂τ 2 − 1 2 ∂φ(x) ∂~x 2 −V[φ(x)] # =i Z τf τi dτ Z ∞ −∞d~x " 1 2 ∂φ(x) ∂τ 2 +1 2 ∂φ(x) ∂~x 2 +V[φ(x)] # =i Z τf τi dτ Z ∞ −∞d~x 1 2∂µφ(x)∂ µ φ(x) +V[φ(x)] , (4.2.3) where now x=xµ= (τ,~x)and ∂µ =

∂ ∂τ, ∂ ∂~x

. We can then define the Euclidean action to be IE[φ(x)] = −iI[φ(x)] t=−iτ = Z τf τi dτ Z ∞ −∞d~x 1 2∂µφ(x)∂ µ φ(x) +V[φ(x)] = Z τf τi dτ Z ∞ −∞d~x q |g(x)| 1 2g µν₍_x₎ ∂µφ(x)∂νφ(x) +V[φ(x)] . (4.2.4)

The propagator changes accordingly to

φ(τ_f,~x)e−(τf−τi)H|φ(τ_i,~x)i = N

Z _φ(_τ_f,~x)=_φ_f(~x)

φ(τi,~x)=φi(~x)

D[φ(x)]e−IE[φ(x)]. (4.2.5)

In the saddle point approximation, the action can now be written as IE[φ(x)] =IE[φcl(x)] + 1 2 Z ddxδφ(x) −∆+V00[φcl(x)] δφ(x) + O(δφ3), (4.2.6)

where the Laplacian gµν

∂µ∂ν = ∂µ∂µ = ∆ = ∂ 2 ∂τ2 +∑ d n=1 ∂ 2 ∂x2n = ∂ 2 τ+∂ 2 ~x. The

Euler-Lagrange equation gives

−_p 1 |g(x)|∂µ q |g(x)|gµν ∂νφcl(x) +V0[φcl(x)] = −∆φcl(x) +V0[φcl(x)] =0, (4.2.7)

so that the propagator becomes φ(τ_f,~x)e−(τf−τi)H|φ(τ_i,~x)i = N Z _φ(_τ_f,~x)=_φ_f(~x) φ(τi,~x)=φi(~x) D[φ(x)]e−IE[φ(x)] = N Z δφ(x)=0∀ x∈∂Ω D[δφ(x)]e−(IE[φcl(x)]+ 1 2 R dd_xδφ₍_x₎₍₋_∆₊_V00_[ φcl(x)])δφ(x)+O(δφ3)) ' Ne−IE[φcl(x)] Z δφ(x)=0∀x∈∂Ω D[δφ(x)]e− 1 2 R dd_xδφ₍_x₎₍₋_∆₊_V00_[_φ cl(x)])δφ(x) ≈e−IE[φcl(x)] (4.2.8)

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4.3 massive free scalar field in d-dimensional euclidean spacetime 32

Replica Trick

Like we saw in the quantum mechanical case, we can also see a correspondence be-tween the time direction and temperature in Euclidean scalar field theory. By imposing periodicity on the Euclidean time, so that 0≤ τ≤β, the propagator changes to

hφ(β,~x)|e−βH|φ(0,~x)i = N

Z _φ(_β,~x)=_φ_f(~x)

φ(0,~x)=φi(~x)

D[φ(x)]e−IE[φ(x)]. (4.2.9)

The propagator is then equal to the density matrix up to a factor of Z hφ(β,~x)|ρ|φ(0,~x)i =φf(β,~x) e−βH Z |φi(0,~x)i = N Z Z _φ(_β,~x)=_φ_f(~x) φ(0,~x)=φi(~x) D[φ(x)]e−IE[φ(x)] ≈ e −IE[φcl(x)] Z (4.2.10)

Znis defined by a trace, where φ(0,~x) =φ(nβ,~x) = ϕ(~x)

Zn=Tr e−nβH = N Z φi(~x)=φf(~x)=ϕ(~x) D[ϕ(~x)] Z φ(nβ,~x)=φf(~x) φ(0,~x)=φi(~x) D[φ(x)]e−IE[φ(x)] ≈ Z φi(~x)=φf(~x)=ϕ(~x) D[ϕ(~x)]e−IE[φcl(x)] (4.2.11)

We can then perform the replica trick by implementing the result for Zn in2.3.18. We

can also compute the replica trick by calculating Tr ρn, which implies that we should first retrieve ρn_{. We set φ}₍_β_,_~_x_{) =}_φ nand φ(β,~x) =φ0, so that hφ(β,~x)|ρn|φ(0,~x)i = hφn|ρn|φ0i = Z n−1

∏

i=1 dφihφn|ρ|φn−1i × · · · × hφ1|ρ|φ0i. (4.2.12)

We can then trace over ρnby setting φn=φ0

Tr ρn=hφn|ρn|φ0i = Z dφ0 Z i−1

∏

n=1 dφihφ0|ρ|φn−1i × · · · × hφ1|ρ|φ0i. (4.2.13)

4.3 m a s s i v e f r e e s c a l a r f i e l d i n d-dimensional euclidean spacetime

In this example we will compute the path integral for a massive free scalar field in d-dimensional Euclidean spacetime and consequently perform the replica trick to compute its entropy. The Euclidean action of a massive free scalar field d-dimensions is

IE[φ(x)] = Z τf τi dτ Z ∞ −∞d~x 1 2∂µφ(x)∂ µ φ(x) + m 2 2 φ(x) 2 = 1 2 Z τf τi dτ Z ∞ −∞d~x ∂µφ(τ,~x)∂ µ φ(τ,~x) +m2φ(τ,~x)2 , (4.3.1)

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4.3 massive free scalar field in d-dimensional euclidean spacetime 33

where d~x=dd−1_{x. In the saddle point approximation, the propagator becomes}

φf(τf,~x) e−(τf−τi)H|φ_i(τ_i,~x)i ' Nexp −1 2 Z τf τi dτ Z ∞ −∞d~x ∂µφcl(τ,~x)∂ µ φcl(τ,~x) +m2φcl(τ,~x)2 × Z δφ(x)=0∀x∈∂Ω D[δφ(x)]exp 1 2 Z ddxδφ(x) ∆−m2 δφ(x) . (4.3.2)

To find φ_cl(x)we need to solve the Euler-Lagrange equation ∆φcl(x) −V0(φcl(x)) = ∆−m2

φcl(x) = ∂2_τ+∂_~2_x−m2φcl(τ,~x) =0. (4.3.3)

We can use the Fourier transform to find a general solution for φcl(x)

φcl(τ,~x) = Z _d_~_p (2π)d−1e i~p·~x φcl(τ,~p) φcl(τ,~p) = Z d~xe−i~p·~xφcl(τ,~x), (4.3.4)

where the momentum vector p= pµ = (p0,~p) = (p0, p1, . . . , pd−1), so that

∆−m2 φ_cl(x) = ∂2_τ+∂_~_x2−m2φ_cl(τ,~x) = ∂2_τ+∂_~2_x−m2 Z _d_~_p (2π)d−1e i~p·~x φcl(τ,~p) = Z _d_~_p (2π)d−1 ∂ 2 τ− ~p 2₋_m2_ei~p·~x φ_cl(τ,~p) =0 → ∂2_τ− ~p2−m2φ_cl(τ,~p) = ∂2_τ−ω_~2_p φ_cl(τ,~p) =0. (4.3.5) Here we defined ω_~2_p = ~p2+m2 (4.3.6) with ω~p= q ~p2₊_m2 . (4.3.7)

We also want the scalar field to be real, so that

φcl(τ,~x) =φ_cl∗(τ,~x) → Z _d_~_p (2π)d−1e i~p·~x φ_cl(τ,~p) = Z _d_~_p (2π)d−1e −i~p·~x φ_cl∗(τ,~p) →φ_cl∗(τ,~p) =φ_cl(τ,−~p). (4.3.8)