Aspects of non-equilibrium QFT: first-order corrections to Stefan-Boltzmann law and particle density

DIPARTIMENTO DI FISICA "E. FERMI"

Master degree course in Fisica Teorica

Master Degree Thesis

Aspects of Non-Equilibrium QFT:

first-order corrections to

Stefan-Boltzmann law and particle

density

Advisor:

Out of equilibrium physics is a branch of modern QFT currently under development. Some powerful formalisms have been developed - the Keldysh one, for example, is extremely useful and widely applied - yet the lack of a good framework and the complexity of the calculations limit the potentiality of these approaches.

This thesis has been written with the hope that some important quantities can be computed even far from equilibrium in a simple manner. The driving idea was to adopt the formalism of quantum field theory to look for corrections to known laws when a small interaction drives a system out of equilibrium. In particular we con-sider two-component systems. Each of the components is in contact with a heat reservoir and the reservoirs are in general at different temperatures and chemical potentials. Switching on an interaction drives the system away from equilibrium and modifies the known equilibrium laws. In this context our new results concern the first nonequilibrium correction to particle density and to the Stefan-Boltzmann law.

This thesis is organized as follows. A concise introduction to the theory is devel-oped in second and third chapters, spanning from definition of a Gibbs state to a formulation of perturbative expansion in the coupling constant when an interaction is present; in the second part (chapters 4 and 5) the reader can find calculations obtained in this work. Along with them, a brief discussion of each result can be found.

Finally, in Chapter 6 are discussed possible future developments and some fields of applicability of this work.

The present job is formulated in natural units where ~ = c = 1, in a spacetime with

d spatial dimensions and signature (+ − ... −).

A spatial vector will be indicated in bold, wheter a spacetime object will be found in italic: p is a d-vector, while p is a d+1-vector and will often be accompanied with a greek (Lorentz) index, pµ_{, to understand its relativistic nature.}

Einstein notation for index summation is used. In some result however we use explicit sum symbols to be clearer. The reader will find, in the second part of the thesis, a lot of long calculations, where being pedantic with notation would lead to a terrible typesetting of equations. Since most of this ugliness comes from exponen-tials, the most obvious solution has been to collapse their notation. Whenever two spacetime coordinates x and y in an exponential are subtracted, the operation will be indicated by ∆xy instead of x − y. Again in exponentials, the standard notation

of implying Lorentz indices is adopted, so that

eipx= eipµxµ_.

As we will be dealing with lots of products of fields, writing down all the integra-tion measures would have rendered this work unreadable, so they will more often than not be understood. The notation will, however, help us in keeping track of these measures: whenever a momentum vanishes from the equations, it means that has been integrated out. At the end of calculations the surviving measure will be inserted back.

Finally, we introduce Dirac gamma matrices in d-dimensions as generalization of the 4-dimensional case. They will be written as γµ_{, with µ = 0, . . . , d. For a}

Summary iii

Notation _iv

I

Introduction

1

1 Overview and motivations ₂

2 Equilibrium theory ₄

2.1 Scalar fields . . . 4

2.1.1 Real scalar field . . . 4

2.1.2 Complex scalar field . . . 5

2.1.3 A− representations . . . 8

2.2 The Dirac field . . . 14

2.2.1 A+ representations . . . 18

2.3 Normal ordering. . . 22

2.4 Noise power . . . 24

2.5 The thermodynamical approach to Stefan-Boltzmann law . . . 24

3 Out of equilibrium theory ₂₆ 3.1 Time evolution . . . 27

3.2 Perturbative expansion . . . 27

3.3 An application: the NESS . . . 28

3.4 Interactions . . . 29

3.4.1 |φ|4 _{interaction . . . 29}

4 Conserved charges out of equilibrium ₃₂

4.1 Scalar fields, energy density . . . 32

4.1.1 Real scalar field at equilibrium: the T4 _{power law . . . 32}

4.1.2 Complex scalar field at equilibrium . . . 34

4.1.3 Driving out of equilibrium: φ4 _{interaction . . . 35}

4.2 Scalar field, U(1) symmetry . . . 38

4.3 Dirac field, energy density . . . 41

4.3.1 Higgs-like Yukawa model . . . 42

4.3.2 First order corrections to the energy . . . 43

4.4 Yukawa model, U(1) symmetry . . . 45

5 Thermal noise ₄₇ 5.1 2-point function for the scalar field current . . . 47

5.1.1 Noise power density. . . 48

5.1.2 2-point function for the Yukawa current . . . 49

5.1.3 Noise power density. . . 50

6 Conclusions ₅₃ Appendices 55 A Noether’s theorem ₅₆ B Expectation values for heated complex scalar fields ₅₈ B.1 Contractions. . . 58

B.2 Normal ordered products . . . 59

B.2.1 Averages of commutators of ordered operators . . . 61

C Expectation values for heated fermionic fields ₆₂ C.1 Contractions. . . 62

C.2 Normal ordered products . . . 63

C.2.1 Averages of commutators of ordered operators . . . 64

C.3 Gamma matrices in d+1 dimensions . . . 65

D Table of integrals and known functions ₆₇

Overview and motivations

Nonequlilibrium physics is fundamental for describing the real world. In this context interactions play a fundamental role: they allow for aggregation, absorbtion, emis-sion, colliemis-sion, exchange of properties. They are what makes our world interesting, what makes it to exit from equilibrium.

In the last decades the number of research activities in areas where quantum field processes of nonequilibrium many-body systems is dominant has grown up a lot. This includes early universe cosmology with high-precision observations (such as WMAP) and simulations – for instance, the primordial density fluctuations that later gave rise to structures like galaxies –, nuclear particle physics in the rela-tivistic heavy ion collision (RHIC) experiments that hope to see the formation and evolution of a novel phase of matter – the quark-gluon plasma –, cold atom con-densation (BEC) in highly controllable environments, BEC/BCS (low-temperature superconductivity) crossover, quantum mesoscopic processes, molecular dynamics and collective phenomena in condensed matter systems [1, 2, 3, 4].

All these researches require concepts from quantum field theory to treat the nonequi-librium dynamics of relativistic many-particle systems and for the understanding of basic issues like dissipation, entropy, fluctuations, noise and decoherence in these systems. QFT in fact grants various fundamental properties: that interactions are local and causal, that energy is bounded, and that relativistic processes can modify particle number. Many formalism have been developed over the years to treat the non equilibrium problem; the reader can find inspiring approaches in [5, 6, 7].

The path taken in this thesis differs a bit from general non-equilibrium problems: it specialises on weakly interacting systems coupled to a thermal bath. Cold atoms trapped in an electromagnetic potential are a common example.

This thesis is organised in the following way: Initially an equilibrium theory in a thermal bath is presented, along with the quantities that will be studied in the second part; then a short review on how to deal with small interactions is presented.

Equilibrium theory

The study of quantum non-equilibrium phenomena is currently under development. Many results have been found, with various formalism, but most of them have a common starting point: quantum field theory. In this chapter we introduce free fields – their lagrangians, energy and number conservation laws – therefore describing an equilibrium theory [8, 9]. To account for thermodynamical quantities (temperature and chemical potential) we introduce an alternative representation for field operator algebra: the Gibbs state [7].

2.1

Scalar fields

A scalar field is an object invariant under spatial rotations. It can be both real or complex and can be used to represent spinless particles such as the Higgs boson [10], the inflaton [11] or some non-elementary particles like atoms, for example.

2.1.1

Real scalar field

The simplest realization of a scalar field is a map from spacetime coordinates to real numbers. Calling ϕ the field, the lagrangian describing its dynamics is

L= 1

2∂µϕ∂µϕ − 1

2m2ϕ2, (2.1)

where m is the mass of ϕ, m > 0. From the previous relation we find the equations of motion δ δϕL= ∂µ δ δ∂µϕ L =⇒ ( + m2_{)ϕ = 0 ,}

where  is the d’Alembert operator ∂µ_∂

µand what we found is the free Klein-Gordon

equation. Its most general solution, imposing a reality condition, is

ϕ(x) = Z ddp (2π)d√2 ω p  ape−ip µ_x µ+ a∗ peip µ_x µ   p0= ωp (2.2) with ω2

p = p2 + m2. From now on we will drop the (d+1)-vector indices in the

exponentials, using the convention that they are present but not written unless dif-ferently specified: eipµ_x

µ = eipx. To refer to spatial coordinates we will use bold

items: pspatial = p. The condition p0 = ωp will also be implied from now on.

Starting from the lagrangian, we can define the conjugate of ϕ as Π = δL

δ∂0ϕ

= ∂0_{ϕ . (2.3)}

In order to quantize the theory, we promote both the field and its conjugate mo-mentum to operators and impose on them the equal time canonical commutation relations (CCR):

[ϕ(x, t), Π(y, t)] = iδ(x − y) . (2.4)

Substituting their expansions, we find that the Fourier coefficients – now promoted to operators and called creation (a†_{) and destruction (a) operators– must satisfy the}

commutation relations: h ap, a†q i = (2π)d_{δ(p − q) , (2.5a)} h ap, aq i =h a†_p, a†_qi= 0 , (2.5b)

which define an algebra A−(a, a†) that we will later study in order to construct the

Gibbs state.

2.1.2

Complex scalar field

Consider now the field

φ = ϕ1√+ iϕ2

2 .

If we write the quantum lagrangian

L= ∂µφ†∂µφ − m2|φ|2, (2.6)

it is easy to check that this is the sum of two lagrangians describing the two inde-pendent fields ϕ1 and ϕ2. In particular the solutions to the equations of motion for

these two independent fields can be written as a superposition of plain waves: φ(x) = Z ddp (2π)d√2 ω p  ape−ipx+ b†pe ipx (2.7) φ†(x) = Z ddp (2π)d√2 ω p  a†_peipx+ bpe−ipx  (2.8) We can now forget about the construction technique we followed and take the pre-vious equations as definitions.

The conjugate momenta for φ and φ† _are:

Π = ∂0_φ† _Π†_{= ∂}0_{φ ,}

so that we can impose the CCRs

[φ(x, t), Π(y, t)] = iδ(x − y) (2.9)

for both φ and φ†_{. Again, substituting, we can move these conditions over a and b:}

h ap, a†q i = (2π)d_{δ(p − q)} h_b p, b†q i = (2π)d_{δ(p − q) (2.10)} with h ap, bq i =h a†_p, b†_qi=hap, b†q i =h a†_p, bq i = 0

so that they define two copies of the same scalar algebra A−. Since all the a and a†

commute with all b and b†_{, these two algebras are independent and disjoint.}

Before studying the scalar algebra, let’s construct the hamiltonian density

H =X

i

Πi∂0φi− L= Π Π†+ ∂iφ†∂iφ+ m2|φ|2,

which can be rewritten in terms of creation and annihilation operators. If we in-troduce Normal ordering1_{, that moves all the daggered operators to the left and}

the others to the right; and substitute the fields definitions, we find for the total hamiltonian: H= Z _dd_{p d}d_q (2π)2d2√_ω pωq _ ωpωq+ piqi  : a†_peipx− bpe−ipx  aqe−iqx− b†qe iqx : + + m2_: a†_peipx+ bpe−ipx  aqe−iqx + b†qe iqx :ddx =Z _(2π)d_2ddp d₂√dq ωpωq   ωpωq+ piqi+ m2  a†_paqei(p−q)x+ b†qbpe−i(p−q)x  + −ωpωq+ piqi− m2  a†_pb†_qei(p+q)x+ aqbpe−i(p+q)x  ddx

integrating over x we get a (2π)d_{δ(p−q) for the first line and a (2π)}d_{δ(p+q) for the}

second. When integrating momenta variables, the second line becomes null and we get: H = Z ddp (2π)dωp  a†_pap+ b†pbp  . (2.11)

Note that in the same manner one can derive the hamiltonian for a real scalar field: H= Z ddp (2π)dωp  a†_pap  (2.12) Conserved quantities

We now move to the study of symmetries of the lagrangian (2.6). Consider a U(1) infinitesimal, global, continuous internal symmetry of the fields, under which

φ(x) → φ0(x) = eiθφ(x) ' φ(x) + iθφ(x) = φ(x) + θδφ(x)

The lagrangian (2.6) is invariant under this transformation. Noether’s theorem states that to each continuous global symmetry of the lagrangian we can associate a conserved quantity, called Noether current jµ_{, in the sense that ∂}

µjµ= 0.

In particular one can demonstrate that this current is given by2

jµ = δL δ∂µφ

δφ .

From the theorem, it follows that to each conserved current we can associate a charge Q which is conserved by the action:

∂0j0 = ∂iji =⇒

d

dtQ= 0 , Q=

Z

ddxj0;

so for our scalar field the quantity

jµ= i∂µφ†φ − φ†∂µφ= i  φ† ↔ ∂µφ  (2.13) is conserved and gives a conserved charge

QU (1) = i Z ddx  φ† ↔ ∂0φ  . 2_{see AppendixA.}

We want to compute this quantity. Anticipating that we need to normal order it and inserting the definitions (2.7), we find

QU (1)= Z ddx Z _dd_pdd_q (2π)2d2√_ω pωq h ωp:  a†_peipx− bpe−ipx   aqe−iqx + b†qe iqx :+ + ωq:  a†_peipx+ bpe−ipx   aqe−iqx− b†qeiqx  :i = Z ddx Z _dd_pdd_q (2π)2d2√_ω pωq h ( ωp− ωq)  a†_pb†_qei(p+q)x− aqbpe−i(p+q)x  + + ( ωp+ ωq)  a†_paqei(p−q)x− b†qbpe−i(p−q)x i

performing the integration over x, we find a (2π)d_{δ(p + q) for the first line – which}

vanishes when we integrate over p or q because of the parity of ωs – and a (2π)d_δ(p−

q) for the second one, that gives

QU (1) = Z ddp (2π)d  a†_pap− b†pbp  . (2.14)

This can be recognized as the difference in number between particles and antiparti-cles. Being a conserved charge, dynamics induced by the action does not change it. Our lagrangian (2.6) is also Poincaré invariant, which means that another current is conserved: the energy-momentum tensor. This is defined in its canonical form as

Tµν = δL δ∂µφi

∂νφi− ηµνL (2.15)

where the index i refers to all fields in the lagrangian.

If we take µ = ν = 0 in its definition, we see that T00 is exactly the hamiltonian

density (or energy density):

T00=    1 2  ∂0ϕ∂0ϕ+ ∂iϕ†∂iϕ+ m2ϕ2 

for a real scalar field;

∂0φ†∂0φ+ ∂iφ†∂iφ+ m2|φ|2 for a complex one.

(2.16)

2.1.3

A

representations

In this section we study some representations of the scalar algebra (2.5).

Let A = {a†_{(p), a(p)} be some operatorial algebra defined by the commutation rules:}

h

Fock space. We define Fock space [8, 12] as the representation of this algebra

over which we can identify a certain state in an orthogonal basis, the vacuum |Ωi , such that if we apply a destruction operator a on it we get a null eigenvalue:

a(p) |Ωi = 0 .

If we normalize the vacuum to hΩ|Ωi = 1, we can construct the entire space from subsequent application of the creation operator on it; for example the state of n particles with the same p is written as

|p(1), ..., p(n)i = (2ωp)n/2



a†(p)n|Ωi = |n(p)i , where ω(p) is a convenient normalization factor.

We thus see that the Fock space F is the direct sum of tensor products of n single-particle Hilbert spaces :

F =

∞

M

n=0

S+H⊗(n),

where S+ is an operator that symmetrizes the combination of Hilbert spaces.

With these definitions in mind, let’s compute the expectation value for the product of two elements of A− on the vacuum:

hΩ| a(p)a(q) |Ωi_F = hΩ| a†(p)a†(q) |Ωi_F = hΩ| a†(p)a(q) |Ωi_F = 0 (2.18a) hΩ| a(p)a†(q) |Ωi_F = hΩ|ha(p), a†(q)i− a†(q)a(p)|Ωi_F = (2π)δ(p − q) . (2.18b) From these follows the 2-point function for a real scalar field in the Fock represen-tation: hΩ| ϕ(x)ϕ(y) |Ωi_F = hΩ| Z ddp ddq (2π)2d2√_ω pωq apa†qe −i(px−qy)_|Ωi F =Z _(2π)dd_dp_{2 ω} p e−ip(x−y). (2.19)

Since there appears a Lorentz-invariant measure, we can make this invariance ex-plicit, paying the price of inserting a δ( ω2

p − p2 − m2)θ( ωp) to keep consistency3.

What we get is then:

hΩ| ϕ(x)ϕ(y) |Ωi_F =

Z dd+1p

(2π)d δ(p

2_{− m}2_{)θ( ω}

p)e−ip(x−y). (2.20)

Now the Lorentz invariance of this correlator is evident. Observe that energy is bounded from below by the Heaviside θ function4_.

Consider finally the polynomial P = a†_(p)a(q)a†_{(r)a(s) and take its expectation}

value on the vacuum: this is clearly 0. The fact that a(p) acting on the vacuum gives 0 can be seen as requesting that we cannot steal particles from the ground state, fact that, of course, is a good description for a vacuum. We will next consider a representation not sharing these last properties with the present one. Let’s see why.

Gibbs representation. In this thesis we are looking for something more suitable

to describe thermal states: the Gibbs representation [7]. We define it as a para-metric vector space G(β, µ) over which a polynomial in the algebra elemenents has expectation value [13]: D P(a, a†)E G = 1 ZTr h e−β(H0−µN )_P(a, a†)i (2.21)

where H0 is the free Hamiltonian describing the particles we are interested in, β

their inverse temperature, N = a†_{a their number operator, and µ their chemical}

potential. The factor 1/Z is the normalization of the Gibbs state and is given by Z = Trhe−β(H0−µN )i _.

We can think of this state as a reservoir: being illimitate, if we give it some properties no physical process can change them, therefore this is always described by fixed parameters. We have precedently found (2.12) that the free hamiltonian can be written as an integral over a†_{a, the number operator. This implies that we can}

define an effective hamiltonian K:

K = H0− µN = Z ddp (2π)d(ωk− µ)a † pap = Z ddp (2π)dpa † pap. (2.22) From this definition, one can easily check that

[ap, K] = pap

h

a†_p, Ki= −pa†p. (2.23) 4_{This is also known as the step function and is defined as}

θ(x) =

(

0 x < 0

In particular the previous relations lead us to the equations :

ape−βK = e−βpe−βKap (2.24a)

a†_pe−βK = eβp_e−βK_a†

p. (2.24b)

Proving these relations is straightforward: define

F(β) = ape−βK (2.25)

and observe that F (0) = ap. Write

∂ ∂β

F(β) = −apK e−βK = −(p+ K)F (β) ,

where in the last equality we have taken advantage of (2.23). This is then a full Cauchy problem:    F0(β) = (p+ K)F (β) F(0) = ap =⇒ F(β) =    F(β) = ape−(p+K)β F(β) = e−(p+K)β_a p

We see that, by definition (2.25), the first case can’t be a solution unless p = 0; the other, instead, is the desired solution. The a† _{relation is obtained in the same way.}

Together with the cyclic property of the trace and a wise use of CCR (2.5), equa-tions (2.24) can be used to deduce the following expectation values over the Gibbs state: hapi_G = D a†_pE G = 0 (2.26a) hapaqi_G = D a†_pa†_qE G = 0 (2.26b) D a†_paq E G = (2π) d δ(p − q) nG_p (2.26c) D apa†q E G = (2π) d δ(p − q) mG_p (2.26d) with nG_p = 1 eβ( ωp−µ)−1, m G p = 1 + n G p . (2.27)

From algebraic rules only, we have found the Bose-Einstein occupation number for particles with energy ωp and chemical potential µ at inverse temperature β, nGp .

Note that to avoid singularities (bosonic condensation) we choose to put µ < 0. In the second part of the thesis however, whenever a result depending on nG _{is found,}

Observe that of the 4 functions we computed in (2.18a), two are zero even in this representation – namely (2.26b) – while the other two have changed; in particular in this representation a |Ωi /= 0. Note however that in the limit β → ∞, equa-tions (2.26c)-(2.26d) reproduce their correspective in the Fock representation.

In analogy with the Fock counterpart, let’s study the 2-point function for a real scalar field: hϕ(x)ϕ(y)i_G = Z ddp ddq (2π)2d2√_ω pωq _D apa†q E Ge −i(px−qy)₊D a†_paq E Ge i(px−qy) =Z ddp (2π)d2 ω p 

e−ip(x−y)+ nG_pe−ip(x−y)+ nG_peip(x−y)

=Z ddp (2π)d2 ω

p



mG_pe−ip(x−y)+ nG_peip(x−y) (2.28)

Some remarks are required here; they highlight a huge difference between the two representations. Since the integrand depends on the thermal distribution n, which in turn depends on p only through ωp, this two-point function is not Lorentz invariant.

Spatial rotations keep constituting a symmetry, however.

Let’s now see if, as in the Fock case, this correlator has bounded energy. Introducing the Lorentz invariant measure in the second line of (2.28), we see that the first term is invariant under boosts and we forget about it. The others however are not:

take Z d_(2π)d+1p_d δ(p2−m2_{) θ( ω}

p)



nG_pe−ip(x−y)+ nG_peip(x−y)

and send pµ_{→ −p}µ _{for second addend only}

=Z d_(2π)d+1p_d δ(p2−m2_{) θ(p}0_)e−ip(x−y) 1 eβ(p0_−µ) −1− 1 eβ(−p0_−µ) −1 

put now µ = 0 for convenience and rearrange the sum =Z d_(2π)d+1p_d δ(p2−m2_{) θ(p}0_)e−ip(x−y) 1 eβ(p0₎ −1 − 1 eβ(−p0₎ −1  =Z dd+1p (2π)d δ(p 2_−m2_{) (p}0_)e−ip(x−y) 1 eβp0 −1 where  is the sign function:

(x) =    +1 x ≥ 0 −1 x < 0 .

Comparing this result with (2.20), it is clear that the sign function changes the physics: in this representation energy isn’t bounded from below! Before panicking, consider the definition of a Gibbs state: β and µ are fixed. Because of this the state represents a thermal bath with a reservoir and, by its very definition, a reservoir has illimitate energy at disposal. Thus a non-bounded hamiltonian is perfectly admit-table in the theory.

Lastly, observe that in the β → ∞ limit we find same the result as in Fock repre-sentation.

Suppose now that we want to compute the expectation value for a polynomial P over the Gibbs state. P will be of the form:

P(a†)N_{, a}M_{= (a}†₎n1_am1_...(a†)nf_amf

with the ni that sum up to N and the mi to M. Then, using commutation

rela-tions (2.5)) and relation (2.24b), we get:

D P(a†)N, aME G = e −β1Trh_a† 1e −βK_(a†₎n1−1_am1_...(a†)nf_amfi= = e−β1Trh_a† 1e −βK_(a†₎n1−1_am1_...(a†)nf_amf−1h_a q, a † 1 i + a† 1aM i = e−β1  (2π)d δ(p₁− q_N)DP(a†)N −1/1, aM −1/ME G+ +D P(a†)N −1, aM −1a†₁aM E G 

Where with aS/T _{we mean the sequence of S a s without the T one. Iterating the}

procedure of commuting our a†

1 with the remaining polynomial in the second term, it is easy to see that we get to

= e−β1  (2π)d M X i δ(p₁− q_i)DP(a†)N −1/1_{, a}M −1/iE G+ +D P(a†)N, aME G 

From this, exactly as for ((2.26)), we deduce the fundamental relation:

D P(a†)N_{, a}ME G = n G 1 (2π) d M X i δ(p₁ − q_i)DP(a†)N −1/1_{, a}M −1/iE G = XM i D a†₁ai E G D P(a†)N −1/1_{, a}M −1/iE G (2.29)

Iterating the procedure it is straightforward to see that when N /= M the expectation value of P is null. From this we deduce the corollary:

Corollary 1 A necessary condition for an average over the Gibbs state to be

non-null is that the number of creation and annihilation operators is the same.

Note that this is also true for the Fock representation.

2.2

The Dirac field

We now focus our attention on the study of Dirac fermions [8, 9]. They are tensor product of two fundamental spinors and can therefore be seen as (d+1)-dimensional objects5; they also manifestly conserve parity [8].

The fermionic lagrangian is written as L= ¯ψi /∂ − mψ = 1 2ψ¯  i~/∂ − m  −  i ~/∂ + m  ψ , (2.30)

where ¯ψ is a shorthand notation for ψ†γ0, /∂ for γµ∂µ, γµ are the (d+1)-dimensional

Dirac gamma matrices6 _{and the arrow over derivatives specifies the direction over}

which we apply them. Note that because of difficulties in the construction of a physically meaningful set of gamma matrices in even spatial dimensions, everything

we will do for fermions will be limited to the case where d is odd.

From the lagrangian we see that fermionic fields satisfy the equations of motion

(i/∂ − m)ψ = 0 ψ¯(i/∂ + m) = 0 (2.31)

which admit as solution both

ψ(x) = Z _dd_p (2π)d√2 ω p X s  us(p) bp,se−ipx+ vs(p) c†p,se ipx   p 0_{= ω} p (2.32) and ¯ψ = ψ†γ0. Here u

s(p) and vs(p) are (d+1)-dimensional spinors, representatives

of the Spin(d) group that is the double cover of SO(d). From (2.31) we find that they satisfy

(/p − m)us(p) = 0 ¯us(p)(/p − m) = 0 (2.33)

(/p + m)vs(p) = 0 ¯vs(p)(/p + m) = 0 ; (2.34)

5_{provided that d is odd.}

thus the following completeness relations: ¯us(p)ur(p) = 2mδrs X s us(p)¯us(p) = /p + m ; ¯vs(p)vr(p) = −2mδrs X s vs(p)¯vs(p) = /p − m , ¯us(p)vs(p) = us(p)¯vs(p) = 0 (2.35) ¯us(p)γ0vr(p) = u†s(p)vr(p) = D 2 ωpδrs ¯vs(p)γ0vr(p) = vs†(p)vr(p) = D 2 ωpδrs

and the Gordon identities

¯us(p)γµus(q) = ¯us(p) [(p + q)µ+ iσµν(p − q)ν] us(q) (2.36)

are deduced. All these relations will come to hand many times.

The quantity D represents the dimension of the Clifford algebra satisfied by Dirac gamma matrices [14] {γµ_{, γ}ν_{}= 2η}µν_{, while}

σµν = i

2[γµ, γν] .

A brief discussion on this topic can be found in AppendixC.3.

Notation: Now that their structure is clear, we change the notation for

compact-ness reasons and write the spinors as up,s instead of us(p).

The Dirac field has conjugate momentum Π = δL

δ∂0ψ

= i ¯ψγ0 = iψ†,

which satisfies a canonical anticommutation relation CACR, −i {ψr(x, t), Π(y, t)} =

n

ψr(x, t), ψs†(y, t)

o

= δ(x − y) δrs, (2.37)

where r, s are spinorial indices. From this we obtain the relations that define the two copies of spin-1

2 fermionic algebra A+: {bp,s, bq,r}= n b†_p,s, b†_q,ro= {cp,s, cq,r}= n c†_p,s, c†_q,ro= 0 n bp,s, b†q,r o =n cp,s, c†q,r o = (2π)d δ(p − q) δsr. (2.38)

which, again, means that particles and antiparticles belong to disjoint identical al-gebras. We report here, without derivation7_{, the normal-ordered Dirac hamiltonian:}

H=X s Z ddp (2π)d ωp  b†_p,sbp,s+ c†p,scp,s  . (2.39)

Conserved quantities By the very same principles that we followed for complex

scalar lagrangian, let’s look for a Noether current and to the energy-momentum tensor for the Dirac one. The lagrangian (2.30) is in fact invariant under the internal

U(1) symmetry

ψ(x) → ψ0(x) = eiQψ(x) ' (1 + iQ)ψ = ψ + Qδψ .

Noether’s theorem then shows us that the quantity

jµ= δL δ~∂µψ

δψ+ δ ¯ψ δL δ ~∂µψ¯

= ¯ψγµψ (2.40)

is a conserved current and its associated charge, QU (1) = Z ddx ¯ψγ0ψ = Z ddx:ψ†ψ: =Z ddx d d_pdd_q (2π)2d2√_ω pωq X s,r :  u†_p,sb†_p,seipx+ v_p,s† cp,se−ipx  × ×uq,rbq,re−iqx+ vq,rc†q,re iqx : =Z _(2π)d_2dd₂pd√dq ωpωq X s,r Z ddxhei(p+q)xu†_p,svq,rb†p,sc † q,r + + e−i(p+q)x v_p,s† uq,rcp,sbq,r+ − e−i(p−q)xv_p,s† vq,rc†q,rcp,s+ + ei(p−q)x_u† p,suq,rb†p,sbq,r i .

Forgetting about the first two which have 0 expectation value (note that normal or-der on fermions gains a − sign for every swapped operator, we refer to Section2.3for a complete explanation), integrating over spatial coordinates and over a momentum, we find QU (1) = Z ddp (2π)d X s  b†_p,sbp,s− c†p,scp,s  ; (2.41)

7_{Its derivation is completely analogous to that of (2.12), exception made for a minus sign that}

arises from normal ordering of the second addend. Some similar calculations are found in the next pages, so the reader can confront with them.

which means that the number of particles minus that of antiparticles is again a con-served quantity. This shouldn’t surprise us: U(1) global symmetry always reflects this property.

Being Poincaré invariant by construction, also the Dirac lagrangian (2.30) shows a conserved energy momentum tensor. For reasons that go beyond the scope of this work, we would like to work with a symmetric tensor (this condition is required by general relativity). Note however that the canonical tensor (2.15) is not symmetric for non-zero spin fields, so we need to find an alternative definition.

The most direct way to obtain a symmetric tensor is to use the Hilbert action for matter [15]: one can write

S =

Z

L√gddx

where g is the determinant of the metrics with signature (+−−−). From this action we can derive a tensor

Tµν _{= 2} δL

δgµν

− gµν_{L (2.42)}

which is always symmetric and gauge-covariant, so it is the best candidate for a good EM tensor.

Another approach, which provides a correction to the canonical result we found, is the Belinfante one [16]. Exploiting the fact that the EM tensor is defined up to a divergence (see A), we can add to it a divergence-less tensor that makes the sum symmetric. In particular a divergence-free 2-tensor can always be written as

Θµν _{= ∂}

αBαµν with Bαµν = −Bµαν

So that

(T + Θ)µν _{= (T + Θ)}νµ

.

A possible realization of Bαµν that satisfies the requirements is

Bαµν = 1

8ψ¯



{γα_{, σ}µν_{}+ {γ}µ_{, σ}να_{} − {γ}ν_{, σ}αµ_}_{ψ .}

The form of this tensor can be simplified if we remember the properties of gamma matrices introduced in2.2and later explained in Appendix C– the anticommutator of two gamma matrices gives

{γµ_{, γ}ν_{}= 2η}µν

–, the equations of motion for a Dirac fermion (2.31) and if we observe that the following identity is true:

Using all these relations, we find that the Belinfante tensor becomes

Bαµν = 1

8ψ {γ¯ α, σµν} ψ and thus the corrected EM tensor is:

Tµν = i 4ψ¯ h γµ∂~ν− ν~∂ _{+ γ}ν ~ ∂µ− µ~∂ i_{ψ . (2.43)}

Turns out that this result is the same that we would have found from the Hilbert action approach (2.42), after imposing the equations of motion (2.31).

2.2.1

A

representations

We proceed here in analogy with 2.1.3. Let A+ = {bs(p), b†s(p)} be some algebra

defined by the anticommutation relations [17]:

n

bs(p), b†r(q)

o

= 2πδ(p − q)δsr (2.44)

Fock space. We define the Fock space as the representation of this algebra over

which we can identify a certain state, the vacuum |Ωi in an orthogonal basis, such that if we apply a destruction operator b on it we get a null eigenvalue

bs(p) |Ωi = 0 .

The vacuum is normalized to hΩ|Ωi = 1. We can construct the entire space from subsequent application of the creation operator on it The Fock space F is the direct sum of tensor products of n single-particle Hilbert spaces:

F =

∞

M

n=0

S−H⊗(n),

where S− is an operator that antisymmetrizes the combination of Hilbert spaces.

Let’s compute the expectation value for the product of two elements of A+ on the

vacuum: hΩ| bs(p)br(q) |ΩiF = hΩ| b † s(p)b † r(q) |ΩiF = hΩ| b † s(p)br(q) |ΩiF = 0 (2.45a) hΩ| bs(p)b†r(q) |ΩiF = hΩ| n bs(p), b†r(q) o + b† r(q)bs(p)  |Ωi_F = 2πδ(p − q)δrs. (2.45b)

From these follows the 2-point function for a Dirac field in the Fock representation, hΩ| ¯ψ(x)ψ(y) |Ωi_F = hΩ| Z ddp ddq (2π)2d2√_ω pωq X s,r  ¯up,suq,rb†p,sbq,rei(px−qy)+ + ¯vp,svq,rcp,sc†q,re −i(px−qy) |Ωi_F = −DmZ _(2π)dd_dp_{2 ω} p e−ip(x−y), (2.46) because hΩ| cp,sc†q,r|ΩiF = hΩ| n cp,s, c†q,r o |Ωi_F − hΩ| c† q,rcp,s|ΩiF = (2π) d_{δ(p − q)δ} sr

Since this is a Lorentz-invariant measure, we can write it as dd+1_{p, paying the price}

of inserting a δ( ω2

p− p2− m2)θ( ωp) to keep consistency8. Thus

hΩ| ¯ψ(x)ψ(y) |Ωi_F = −Dm

Z dd+1p

(2π)d δ(p

2_{− m}2_{)θ( ω}

p)e−ip(x−y)

and again we see that the Fock representation provides a lower bound for energy.

Gibbs representation. Let’s consider the Gibbs representation [7]. We construct

it with the requirement that the expectation value of a polynomial in the algebra elemenents is D P(bs, b†r) E G = 1 ZTr h e−β(H0−µN )_P(b s, b†r) i (2.47) where H0 is the free Dirac hamiltonian (2.39), β the inverse temperature, N =

R

dpP

sb†s(p)bs(p) their number operator, and µ the chemical potential.

Since the free hamiltonian (2.39) can be written as an integral over b†

s(p)bs(p), which

is exactly the number operator, we can define an effective hamiltonian K:

K = H0− µN = X s Z ddp (2π)d(ωk− µ) b † p,sbp,s = X s Z ddp (2π)dpb † p,sbp,s (2.48)

From this definition, one can easily check that {bp,s, K}= pbp,s

n

b†_p,s, Ko= −pb†p,s. (2.49)

In particular the previous relations lead us to9 _:

bp,se−βK = e−βpe−βKbp,s (2.50a)

b†_p,se−βK = eβp_e−βK_b†

p,s. (2.50b)

8_{see AppendixD}

9_{proving them requires the same steps of scalar case, provided that we use anticommutator}

Together with the cyclic property of the trace and wise use of CACR (2.2), they can be used to deduce the following expectation values on the Gibbs state:

hbpiG = D b†_pE= 0 (2.51a) hbpbqiG = D b†_pb†_qE= 0 (2.51b) D b†_pbq E G = (2π) d δ(p − q) f_pG (2.51c) D bpb†q E G = (2π) d δ(p − q) gG_p (2.51d) with f_pG = 1 eβ( ωp−µ)+ 1 (2.52)

which is the Fermi-Dirac weight function, and

gG_p = 1 − f_pG = 1 − 1

eβ( ωp−µ) + 1. (2.53)

Note that of the 4 functions we computed in (2.45), two are zero even in this repre-sentation, namely (2.45a), while the other two changed; in particular in this repre-sentation b |Ωi /= 0. Observe also that in the limit β → ∞, equations (2.51c)-(2.51d) reproduce their correspective in the Fock representation.

In analogy with what we did in the Fock counterpart, let’s study the 2-point function for the Dirac field:

D¯ ψ(x)ψ(y)E G = Z ddp ddq (2π)2d2√_ω pωq X s,r  D b†_p,sbq,r E G+ ¯up,s uq,rei(px−qy)+ +D cp,sc†q,r E G−¯vp,s vq,re−i(px−qy)  = −DmZ _(2π)dd_dp_{2 ω} p  e−ip(x−y)− fG− p e −ip(x−y)_{− f}G+ p e ip(x−y) = −DmZ ddp (2π)d2 ω p  gG− p e −ip(x−y)_{− f}G+ p eip(x−y)  (2.54) The same remarks we made for the scalar case are valid here. In the limit β → ∞, this correlator – which is not Lorentz-invariant because of its dependence on ωp = p0

only in the weight functions fp = (exp [β( ωp− µ)] + 1)−1 – becomes equal to the

one in Fock representation.

Continuing the parallelism with the scalar case, we can write (2.54) with a Lorentz invariant measure. When sending pµ _{→ −p}µ _{(and forgetting of chemical potential}

for physical considerations only), we find: −Dm Z dd+1p (2π)d δ(p 2_{− m}2_)θ(p0₎ 1 e−βp0 + 1e−ip(x−y)− 1 eβp0 + 1eip(x−y)  → −Dm Z dd+1p (2π)d δ(p 2_{− m}2_)(p0₎ 1 e−βp0 + 1e −ip(x−y) .

We see again that energy is not bounded from below. As we mentioned before, this is a welcome aspect of our theory: if a reservoir is present, energy can be unbounded. We now try to find a rule for rapidly computing expectations values.

Given a polynomial Q in b and b†_,

Q(b†)N, bM= (b†)n1_bm1_...(b†)nf_bmf

with the ni that sum up to N and the mi to M. Then, using the commutation

relations ((C.1b)) and ((2.24b)), we get:

D Q(b†)N_{, b}ME G = e −βMTrh_e−βK_b M(b†)n1bm1...(b†)nfbmf−1 i = = e−βMTrh_e−βKn_b M, a † 1 o − b†₁bM  (b†₎n1−1_bm1_...(b†)nf_bmf−1i = e−β1  (2π)d_{δ(1 − M)}D Q(b†₎N −1/1_{, b}M −1/ME G+ −Db†₁bMQ  (b†₎N −1_{, b}M −1E G 

As in the previous section, bS/T _{indicates the sequence of S b s without the T one.}

Iterating the procedure of anticommuting our bM with the remaining polynomial in the second term, it is easy to see that we get to:

= e−βM  (2π)d M X i (−1)(j−1) δ(i − M)DQ(b†)N/i, bM/ME G+ −DQ(b†)N_{, b}ME G 

where j is the number of all b and b†_{which have been anticommuted with b}

M(which is the number of adjacent permutations made to bring our bMto its current position) and obviously goes from 1 to N +M. Since to get a non-zero result we need N = M, to recover the original polynomial we have to make an odd number of permutations; fact that allows us to always get the correct weight factor for the statitics.

From this, exactly as for ((2.26)), we deduce the fundamental relation: D Q(b†)N, bME G = f G M(2π) d M X i (−1)j δ(i − M)DQ(b†)N −1/i, bM −1/ME G = XM i (−1)jD b†_ibM E G D P(b†)N −1/i, bM −1/ME G (2.55)

Remark: we have to bring together the operators we want to contract. When

we do that, we have to pay a minus sign for each time we flip the first operator with another we are not interested in.

From now on we will only work in the Gibbs representation: every expectation value will be computed on a particular Gibbs state. Whenever we find different algebras, we can separate them as we did when calculating the 2-point function for the Dirac field and treat the two components independently.

2.3

Normal ordering

Now that we have understood most of the basic tools we adopt in this work, it is time to explain normal ordering [8].

Let’s consider a free real scalar field and compute its mean energy. We already found the algebraic form of T00 _{in (2.16), which is nothing but the hamiltonian density:}

E(x) = 1 2  ∂0ϕ∂0ϕ+ ∂iϕ†∂iϕ+ m2ϕ2  .

Its expectation value on the Gibbs state is: hE(x)i = 1 2 Z ddp ddq (2π)2d2√_ω pωq  ωpωq+ piqi+ m2  × ×Da†_paq E ei(p−q)x+Dapa†q E e−i(p−q)x = 1₂Z _(2π)d_dd₂p d√dq ωpωq  ωpωq+ piqi+ m2  × ×npei(p−q)x+ (1 + np)e−i(p−q)x  δ(p − q) = 1₂Z _(2π)ddp_dωp(1 + 2np) .

Although in the massless case ( ω2

p = p2)

hEi= 1 2

Z _dd_p

the integral gives 0, when the mass is non-zero we get into trouble. The second addend, namely R ddp

(2π)dωpnp, is always convergent10, the first one instead reads

1 2

Z ddp

(2π)d ωp

and this is always divergent. Cause of the problem is the aa†_{term. Its presence, that}

has no correspondence in classical theory, is due to the noncommuting nature of the quantum theory. Defining a correct ordering (normal ordering) cures the problem and makes energy well defined.

Let’s see in detail what normal ordering means. Since D

a†aE is finite, we make use

of the commutation relations (2.5) to writeD

aa†E=D1 + a†aE, then throw away the

constant term – responsible of the divergence – and keep the other one, so that:

D

:aa†_:E

=D

a†aE .

If we are instead dealing with a polynomial in the ladder operators, normal ordering works as follows: if the operator on the left is a creation one, it gets out of normal ordering to the left; otherwise it gets commuted with all the operators on its right (no commutator is however inserted) and at the end it gets out to the right:

: a† 1a1...afa†g: = a † 1: a1...afa†g : : a1a † 1...afa†g: = : a † 1...afa†g: a1 (2.56) This procedure works well for bosons. For fermions we must remember that they satisfy canonical anticommutation relations (2.2), so that D

bb†E = D1 − b†bE. This

has implications in the procedure of reordering: whenever a creation operator en-counters a destruction one - and viceversa - they are exchanged and the sequence gets multiplied by −1. What we found in (2.56) is therefore modified to

: b† 1b1...bfb†g: = b † 1: b1...bfb†g: : b1b † 1...bfb†g: = (−1) g_{: b}† 1...bfb†g: b1 (2.57) All this procedure may seem obscure but it reflects the illness in definition of cor-relation functions of distributions at the same point. Normal order, indeed, is a point-like operation and it loses meaning when applied to different space-time points.

2.4

Noise power

Noise power for a quantity A is defined as

P(ν) =

Z ∞

−∞

dτ e−iντhA(x; t)A(x; t + τ)i . (2.58) We are interested in the noise generated by particle flows h:j(x)::j(y):i, where j is the particle number density, conserved when no interaction is present. In particular, since we will do perturbation theory at sufficient low energy, we consider the zero-frequency limit of the power spectrum [17]:

lim

ν→0P(ν) = limν→0

Z ∞

−∞dτ e

−iντ_{h:j(x; t)::j(y; t + τ):i . (2.59)}

This is a crucial quantity, because it allows us to confront our results with real world measurement: if we are able to measure energy fluctuations, we can test our theory [18, 19, 20].

2.5

The thermodynamical approach to

Stefan-Boltzmann law

We now open a parenthesis to show how quantum statistical mechanics deals with the equilibrium problem. Our goal is to derive the equilibrium Stefan-Boltzmann law. As well explained in [12], one usually starts introducing the Fock space F, in order to improve calculability with respect to an Hilbert space H.

If the hamiltonian ˆH of the system commutes with the number operator ˆN, one can

write ˆK = ˆH − µ ˆN = P α  ωαnα − µnα  , with P

αnα = N. The grand partition

function becomes Q= Trhe−β ˆKi H = Y α X nα hnα|e−β(ωαnα−µnα)|nαi ,

where α represents all possible states of the particles. For fermions we have that the only possible occupation numbers are nα = 0,1, so that

1

X

nα=0

e−β(ωαnα−µnα) = 1 + e−β(ωα−µ);

for bosons instead, provided that we choose a negative chemical potential µ < 0, we

find _∞

X

nα=0

e−β(ωαnα−µnα) = 1

As a consequence then write Q=    Q α(1 + e−β(ωα−µ)) for fermions (F); Q α(1 − e−β(ωα−µ))−1 for bosons (B).

Since the grand potential is defined as Ω = −1

β ln Q, and the number of particles as

N = −∂µΩ = ± 1 β X α ∂µln h 1 ± exp − β(ωα− µ) i ,

we find for occupation numbers the Bose-Einstein and Fermi-Dirac distributions:

nα,F =

1

eβ(ωα−µ) + 1; nα,B =

1

eβ(ωα−µ)−1.

Remember now that the sum over α is a sum over spin states and momenta, therefore the total energy of a system composed by non-interacting bosons or fermions is

hEi= X spin Z ddp (2π)dωpnp =            Z ddp (2π)dωpn B

p for spin-0 Bosons;

2Z ddp (2π)dωpn

F

p for spin-12 Fermions.

(2.60)

If in both cases we assume a massless theory and set the chemical potential to 011_,

we obtain the Stefan-Boltzmann law for free spin-0 and spin-1

2 particles: hEi=          π2 60β

−4 _{for spin-0 Bosons;}

7 8

π2

30β−4 for spin-12 Fermions.

.

11_{Note that while with µ = 0 the convergence criterion for the BE distribution does not work,}

energy can be derived from the free energy F = −β−1P

αln(1 ± e

Out of equilibrium theory

What we have said up to now is valid for equilibrium theory. What happens when we take fields in different Gibbs states and allow them to interact? A quantitative an-swer to the question, for the two models we will study, is what we seek in this thesis.

R1

R2

R1

R2

`

Figure 3.1. Two reservoirs: (a) dis-joint, each has its own quantities; (b) allowed to exchange quantities through a link `. The circle represents interaction, schematized to be in the intermediate region only,

The general non equilibrium problem is formulated as follows: take N reservoirs Ri, each characterized by a value bi of a

certain quantity B, with i = 1...N. Create now, for each j, k in the sys-tem, some link `j,k between the

reser-voirs Rj and Rk, so that they can

com-municate and modify their value of B. Suppose also that the flux that passes through the link, Φ(bj, bk; `jk), can

as-sume a maximum value ¯Φ(bj, bk; `jk), so

that dbi

dt  bi ∀i (this effectively says

that Rs are reservoirs).

Our apporach is to consider two fields in different Gibbs states, both de-fined over all space-time, and schema-tize the link through the coupling con-stant g. Put in this way, a general non-equilibrium field theory problem can be seen as figure3.1.

3.1

Time evolution

Let’s start our approach to out of equilibrium theory in pursue of a formula able to describe time evolution for operators[21]. In the Schrödinger picture states can evolve over time and operators are always the same, so we can write ρ = ρ(t). Sup-pose we know the initial state of our system, ρ0; suppose also that this describes

an equilibrium condition. We know that time evolution is described by an antiuni-tary operator U(t, t0) = T



exp [−iRt0

t0 dt

0_H(t0_)]_{, where T represents time-ordering.}

Time evolution for the density matrix is written as [22]

ρ(t) = U(t, t0)ρ(t0)U†(t, t0)

and every other operator describing an observable is time-independent. Thus the time evolution for A is given by:

hAi_t= 1 ZTr [ρ(t)AS] = 1 ZTr [U(t, t0)ρ(t0)U(t, t0)AS] = 1 ZTr [ρ(t0)AH(t)] (3.1) where in the last equality we made use of cyclicity of the trace and exploited the correspondence between Schödinger and Heisenberg pictures to move the time de-pendency from ρ to A.

Note that we need to normalize the result in order to get a physical result, with normalization being the sum af all possible states: Z = Tr [ρ(t0)].

Both these approaches, though, are not very useful in real computation. In order to improve our condition, we introduce the interaction picture I. Here both states and operators evolve in time following

|ψI(t)i = eiH0t|ψS(t)i AI(t) = eiH0tASe−iH0t.

As a consequence,

hAi_I,t= 1

ZTr [ρ(t)IAI(t)] = hAiS,t , (3.2) so that the expectation values we find within this picture don’t need to get trans-formed into the others.

3.2

Perturbative expansion

A complete solution to (3.2) is however hard to find in general; for this reason we will try a perturbative approach [21]. Starting from the Heisenberg picture, where AH(t) = U(t, t0)AH(t0)U†(t, t0), suppose we can write H = H0+ HI, with H0 the

free hamiltonian and HI an interaction turned on adiabatically that depends upon a

representation and assume that the interaction hamiltonian does not depend on time (this physically means that the adiabatical process responsible of its introduction is small and we can write HI(t) = HI(1 − e−ηt) and take 1/η little with respect to the

typical time over which we observe the system, so that HI(t) ' HI): We can write

a Schrödinger equation for the time evolution operator [9]:

i∂

∂tU(t, t0) = HI(t)U(t, t0)

Imposing U(t, t) = 1 one can demonstrate that U(t, t0) = e Rt t0iH0e −Rt t0iH_e Rt0 t0iH0_.

Now we can write hAi in the Heisenberg picture in terms of time evolution in the interaction one: hA(t)i = 1 ZTr " ρ(t0)e −Rt0 t0iH0_e Rt t0iH_e− Rt t0iH0A(t₀) e Rt t0iH0e− Rt t0iH_e Rt0 t0iH0 # = 1 ZTr  ρ(t0)e Rt t0iHA_I(t)e− Rt t0iH 

Expanding the exponentials, noting that we can again get rid of H0 terms in H by

cyclicity of the trace and omitting from now on the representation we are considering (hint: the interaction one), we find the fundamental formula [23]:

hA(τ)i = 1 ZTr ( ρ0 A(τ) − i Z τ −∞[A(τ), HI(t)] dt+ + (−i)2Z τ −∞dt Z t −∞dt1[[A(τ), HI(t)] , HI(t1)] + . . . !) (3.3) This is the main formula we will use during our calculations. Since we are interested in first order corrections, the first line will suffice to do the job.

3.3

An application: the NESS

A non-equilibrium steady state (NESS for short) is a particular regime that gets instaurated between different subsystems in a non-equilibrium condition. As the name suggests, it is steady (or stationary), so we expect quantities computed in this regime to be time-translationally invariant [24, 17].

Since we are not in equilibrium, this condition implies either that all exchanged quantities are null or that the sources are approximately infinite – which means that they are reservoirs –. In the latter case we recognize the usefulness of a de-scription in terms of Gibbs states.

Tank 1

Tank 2

Figure 3.2. Two big containers con-nected through a small pipe. After some time, the flow in the pipe is con-stant and the quantity of water in the containers changes slowly.

There are plenty of NESS examples in everyday world. One of the simplest is the following: take two very large containers and fill them up with wa-ter, then put them at different heights. The water inside will acquire poten-tial energy, U1 > U2. Connect

the two containers through a small pipe, in a fashion that the water can flow between them, but in lit-tle quantity. After some time re-quired by water to fill the pipe and reach its stationary speed, we have a NESS.

We won’t limit ourselves to the study of NESS, yet within this regime our theory behaves well and if the coupling is sufficiently small our first order approximation gives valid results even if when gt  1. In particular the formalism we developed can be applied to general NESS in a direct manner, without any modification. The interested reader can find cross-check our results with those present in [17].

3.4

Interactions

Now that we have developed a perturbative theory to deal with out of equilibrium systems, it is time to talk about the models we are taking under examination in the thesis. First of all we need at least two subsystems described by different Gibbs states; in other words we need more than one field. Then we need a way to couple these states: an interaction. If this interaction is small enough we can use pertur-bation theory and apply (3.3).

3.4.1

|φ|

_interaction

Take two real scalar fields ϕA, ϕB living in the different Gibbs states A and B. If

we insert a coupling term in the lagrangian, we expect these two fields to mix their original properties in some unspecified way. A good interaction term is g(ϕAϕB)2,

where we choose to square in order to get an adimensional coupling constant and non-zero expectation values at first order. The lagrangian describing such system is

L= L0− LI = ∂µφ†∂µφ − m2|φ|2−

g

3.4.2

Yukawa interaction

Another interesting possibility is the coupling of some fermions ψ to a scalar ϕ in what is called a Yukawa interaction:

LI = gϕ ¯ψψ .

This theory, which was first introduced by Yukawa to study nucleon couplings, is widely adopted to describe low-dimensional interactions of phonons with electrons in metals, or to implement the Higgs mechanism to give mass to fermions. In particular we will take a real scalar field and put it in a certain Gibbs state Gϕ, then take a

Dirac field and put it in another Gψ. Since calculations for this case turn out to be

complicated, we give the scalar a vev (vacuum expectation value) v on Gϕ such that,

instead of zero, we find

hϕi_G

ϕ = v .

Given this assumption, it is possible that fermions will see it as a background and won’t interact with it – at least at first order, whereas at second order we think it would give many contributions –, but this needs to be checked.

Conserved charges out of

equilibrium

This part is completely dedicated to our central objective: finding the out of equi-librium corrections to the particle number and to the Stefan-Boltzmann law. The present chapter is divided in different sections, each dedicated both to the scalar |φ|4 and the Yukawa ϕ ¯_ψψ models.

For the first model we consider different conditions for the scalar particle and for its antiparticle, thus attributing different temperatures and chemical potentials to φ and to φ†_{. For the second, instead, fermions and antifermions are in the same}

state but the scalar field is set at different temperature and chemical potential.

4.1

Scalar fields, energy density

It is a good idea to warm up with a little example before starting with our real job. Let’s first take a free real scalar field and compute its Stefan-Boltzmann law, just to make some experience with the calculations and the formalism we have introduced. We will then move on with a complex scalar field and study also the interaction regime.

4.1.1

Real scalar field at equilibrium: the T

power law

Suppose we have a non-interacting real scalar field ϕ, heated to an inverse temper-ature β and with chemical potential µ. Its lagrangian reads:

L= 1

2∂µϕ∂µϕ −

1

We are interested in the energy density of this field, which is just the 00 component of the energy-momentum tensor:

Tµν = δL δ∂µϕ ∂νϕ − ηµνL = ∂µ_ϕ∂ν_{ϕ −} 1 2ηµν∂ρϕ∂ρϕ+ 1 2ηµνm2ϕ2 (4.2) Taking the expectation value on the Gibbs state of the 00 component gives us the en-ergy density of the field. In doing this calculation we have to be careful: an operator defined at a single point must be regularized to avoid ill-defined expectation values which give infinities. This is easily done Normal-ordering the operator (see2.3). Remembering (2.2) and (2.26), we find:

D :T00_(x):E₌ 1 2 D :∂0_ϕ∂0_{ϕ+ δ} ij∂iϕ∂jϕ+ m2ϕ2: E =Z _(2π)d_2ddp d₂√dq ωpωq ωpωq+ p · q + m2 2 D a†_qap E +D a†_paq E =Z _(2π)ddp_d ωpnGp = hEi (4.3)

because the averages give us a (2π)d_{δ(p−q) term that we have integrated. What we}

found is of course the expectation value of the hamiltonian we computed in (2.12). Here nG

p is the bosonic weight function defined in (2.27) as:

nG_p = 1

eβ( ωp−µ)−1 , _β→∞lim n

G

p = 0

and µ is the chemical potential of our field. We can compute analitically our (4.3) integral in the particular case of a massless field: if m = 0 in fact, ωp is just the

modulus of p and the integral becomes rotationally invariant and doable (see the table of integrals in AppendixD). In particular, using z = eβµ _{we get (see D.3):}

hEi= 2π d 2 (2π)dΓ(d 2) Z ∞ 0 dp p d z−1_eβp₋1 = 2 (4π)d/2Γ(d 2) β d+1 Z ∞ 0 dp p d z−1_ep₋1 = 2 (4π)d/2Γ(d 2) βd+1 Γ(d + 1) Lid+1(z) (4.4)

In the case of µ = 0 the polylogarithm1 _{reduces to Li}

t(1) = ζ(t), so for d > 1 we

find the easier relation:

hEi= 2

(4π)d/2Γ(d

2) β

d+1Γ(d + 1) ζ(d + 1)