(p.435) Appendix F Quantum Fields
(p.435) Appendix F Quantum Fields
(p.435) Appendix F
Quantum Fields
F.1 Classical Oscillator
F.1.1 Lagrangian formalism
The action for a classical oscillator of mass m and frequency ω is
The variation of the action with the fixed end points gives the equation of motion
A solution of this equation for the initial conditions q(0) = q _{0}, q̇(0) = q̇_{0} is
In the comparison of the quantum and classical theory an important role is played by the action as a function of its end points. Suppose the initial point at the time t _{1} is q _{1} and the final point at the time t _{2} is q _{2}. The value of the action calculated for the solution Eq. (F.1.3) obeying these boundary conditions is
where Δt = t _{2} − t _{1}.
F.1.2 Hamiltonian formalism
Denote
then the Hamiltonian of the system is
(p.436) In this formalism {p, q} are considered as coordinates in a 2D phase space. For any two functions on the phase space, A = A(p, q) and B = B(p, q), one defines the Poisson brackets
It is easy to check that
The oscillator equations in the Hamiltonian form are
This system is equivalent to Eq. (F.1.2) and its solution, with the initial data (p _{0}, q _{0}), is
F.1.3 Euclidean oscillator
In the quantum theory the Euclidean version of quantum oscillators is quite often used. In particular, this allows one to establish general relations between quantum observables in the thermal state and the Euclidean expectation values. To perform a transformation to the Euclidean space we use the following Wick's rotation of time t
After making this analytical continuation one considers τ as a real parameter, or the Euclidean time. We define the action for the Euclidean oscillator as follows:
The variation of this action with the fixed end points gives the equation
A solution for the ‘initial’ data $q({\tau}_{0})={q}_{0},\text{}\frac{dq}{d\tau}({\tau}_{0})={\dot{q}}_{0}$ is
The Euclidean action as a function of end points is
Here, Δτ = τ _{2} − τ _{1}.
(p.437) F.2 Quantum Oscillator
F.2.1 Heisenberg picture
For a quantum oscillator, q and p are operators obeying the following canonical commutation relations
The quantum Hamiltonian Ĥ is defined as
In the Heisenberg picture q̂ and p̂ are time‐dependent operators and their evolution is determined by the following equations of motion:
These relations imply that any operator Â that is a function of q̂ and p̂ obeys a similar equation
Since Ĥ does not depend on time one can write a solution of this equation in the form
It is convenient to rescale q and p
Working with the new variables q̃ and p̃ is equivalent to taking the mass parameter m equal to 1. From now on we shall use the rescaled variables and omit the tilde. In what follows we shall also put the Planck constant ℏ equal to 1.
F.2.2 Operators of creation and annihilation
Let us write the operators q̂(t) and p̂(t) in the form
Since q̂ and p̂ are Hermitian operators the operator â† is the Hermitian conjugated operator to â. Using Eq. (F.1.10) and Eq. (F.2.5) one gets
Here, constant operators â and â† are the corresponding initial data. These operators obey the following commutation relations:
(p.438) They are known as the operator of creation, â †, and the operator of annihilation, â. The Hamiltonian Ĥ of the quantum oscillator, written in terms of the operators of creation and annihilation, takes the form
In the occupation number representation the basis in the Hilbert space of states is
This basis is orthonormal
One also has
The state ǀ0⟩ has the lowest possible energy ω/2. It is called a ground state or a vacuum.
F.2.3 Thermal quantum oscillator
Let Θ be a temperature. We denote by Θ = 1/Θ the inverse temperature of the system. A thermal state in quantum mechanics is described by a thermal density matrix
The partition function Z(β) is determined by the normalization condition
that is by the following expression
The partition function for the quantum oscillator can be easily calculated using the fact that ǀn⟩ are eigenvectors of the Hamiltonian Ĥ with the eigenvalue equal to (n + 1/2)ω. One has
The partition function determines the free energy F defined by the relation
All the thermodynamical characteristics of the canonical ensemble can be obtained from its free energy. For a thermal quantum oscillator the free energy is
(p.439) The temperature ‐independent term ω/2 in the free energy is a result of the presence of the vacuum energy ω/2 term in the expression for the Hamiltonian Eq. (F.2.10). We can rewrite Ĥ in the form
We use the following standard notation for the thermal average of an operator Â
The expression for the thermal average can also be written in the form
F.2.4 Green functions
There is a variety of different Green functions or two‐point correlators, which proved to be useful in quantum physics. Usually, they have a form of the average value of bilinear combinations of q̂(t) and q̂(t′). They differ by the choice of the state and the concrete form of the bilinear form, which may involve, for example, special ordering of the operators. We focus now our attention on the following thermal correlators
Let us calculate ${\mathcal{G}}_{\beta}^{+}(t,{t}^{\prime})$ One has
Using Eq. (F.2.7), Eqs. (F.2.8), and (F.2.13) one obtains
It is easy to check that
Using Eq. (F.2.25) and Eq. (F.2.26) one obtains
(p.440) In the zero‐temperature limit, when β → ∞, the second term in the right‐hand side vanishes. Thus^{1}
From the representation Eq. (F.2.27) it is evident that ${\mathcal{G}}_{\beta}^{+}(t,{t}^{\prime})$depends only on the time difference t − t′ and, hence, it is a function of only one variable. We denote this function by the same symbol ${\mathcal{G}}_{\beta}^{+}(t)$ so that one has
This property is generic for any two‐time thermal correlator ⟨Â(t)B̂(t′)⟩_{β}. In order to prove this we use the property that for any bounded operators Â and B̂ one has
Let us substitute the following expressions
into the average ⟨Â(t)B̂(t′)⟩_{β}. Then, we apply Eq. (F.2.30) and the property that e ^{− βĤ} commutes with Û_{t} and ${\widehat{U}}_{t}^{1}$ As a result of simple manipulations we obtain
One can use the expression Eq. (F.2.29) for the functions ${\mathcal{G}}_{\beta}^{+}(t)$ and ${\mathcal{G}}_{\beta}^{}(t)$ to define them in the complex plane of the variable t. Then, it is easy to check that the following relation is valid
This result is known as the Kubo—Martin—Schwinger relation, or briefly the KMS relation. Similar KMS relations are valid for any thermal two‐point correlator
To prove this result we use relations Eq. (F.2.31) for a complex time t + iβ to obtain
(p.441) so that
Starting with ${\mathcal{G}}_{\beta}^{\pm}(t)$ functions one can construct other Green functions for the quantum oscillator in the thermal state. In particular, for the commutator [q̂(t), q̂(t′)] one has
This correlator in fact does not depend on the temperature, and it coincides with the vacuum expectation value. The reason is evident: The commutator [q̂(t), q̂(t′)] is a c‐number. A similar thermal Green function with the anticommutator {…,…} being substituted for the commutator is known as the Hadamard function^{2}
Another Green function that plays an important role in the quantum theory is the causal Green function or the Feynman propagator. It is defined as follows
where the T‐product of the operators q̂(t) and q ̂(0) is defined as
Here, θ(t) is a Heaviside step function. The definition Eq. (F.2.39) implies that
This function is continuous at t = 0, but its derivative has a jump. 𝔖_{f,β} can be defined as a solution of the following equation
It obeys the following boundary conditions: At zero temperature this function propagates positive frequency modes to the future and negative frequency modes to the past.
Equation (F.2.35) can be used to determine the evolution of the operators in the purely imaginary or Euclidean time
(p.442) The following object is known as the Matsubara propagator
The operator T _{τ} is the Euclidean time ordering
One has
For τ ϵ [0,β] the Matsubara propagator Δ(τ) coincides with ${\mathcal{G}}_{\beta}^{+}(i\tau )$ and is of the form
One can easily check that
In other words, the Matsubara propagator is periodic in the imaginary time with the period β. It can also be uniquely defined as the Green function for the Euclidean oscillator obeying the differential equation on a circle of length β
Here, δ _{β} (τ) is a delta‐function defined on a circle with the period β
F.3 Quantum Field in Flat Spacetime
F.3.1 Classical scalar field
As an example of the field theory we consider a scalar massive field ϕ in a flat (3 + 1)‐ dimensional spacetime. The action is
Here, t = T and X are Cartesian coordinates, and m is the mass of the field. The variation of the action gives the field equation. For the scalar field it is the Klein‐Gordon equation
(p.443) It is convenient to consider a system in a box with the sizes {L _{1}, L _{2},L _{3}} and to impose the following boundary conditions
Momentum π conjugated to the field ϕ is
The Hamiltonian of the system is
Denote
where n _{i} are positive integer numbers. Then, a complete set of eigenfunctions of the Laplace operator Δ with the imposed boundary conditions is formed by functions
where V = L _{1} L _{2} L _{3} is the volume of the box. These functions obey the equation
and the normalization condition
A general solution of Eq. (F.3.2) can be written as
where the amplitudes of the field q _{k} (t) satisfy the equation
Thus, the amplitude of the field for a given wave number k obeys the oscillator equation. The reduction of the field theory Eq. (F.3.1) to the infinite set of oscillators Eq. (F.3.11) enumerated by k can be made more transparent by substituting the field decomposition in harmonics Eq. (F.3.10) into the Lagrangian Eq. (F.3.1). This substitution gives the following result
(p.444) F.3.2 Quantum field
To quantize the field ϕ it is sufficient to quantize a decoupled set of oscillators described by the Lagrangian Eq. (F.3.12). To do this one considers the amplitude q _{k} and a conjugated momenta p _{k} = q̇_{k} as Hermitian operators obeying the canonical commutation relations
Using the completeness of the orthonormal basis Eq. (F.3.7) it is possible to show that these relations are equivalent to following canonical commutation relations for the field variables
One may describe the quantum field as an infinite set of oscillators. But now each oscillator is ‘living’ not in the ‘physical space’, but in a ‘space of amplitudes’. Spatial modes Φ _{k} describe an amplitude of the probability for a given mode to be in the vicinity of the space point X. The quantum nature of the field is connected with discrete levels of its quantum amplitudes. Since the free quantum field is equivalent to a set of decoupled oscillators, one can easily apply all previous results concerning the quantum oscillator to such a set. In particular, the operators of creation and annihilation for each mode k are related to the amplitudes q̂_{k} and their momenta q̂_{k} = ${\widehat{p}}_{\text{k}}={\dot{\widehat{q}}}_{k}$as follows
where
The operators a ̂_{k} and ${\widehat{a}}_{k}^{\u2020}$obey the commutation relations
The Hamiltonian Eq. (F.3.5) written in terms of the operators of creation and annihilation takes the form
A vacuum is a state ǀ0⟩ with the lowest possible energy. It is defined by the condition
The quantity E _{0} is the energy of the vacuum zero‐point fluctuations. The vacuum energy E _{0} is unobservable. However, in the presence of an external field and/or for non‐trivial boundary conditions the vacuum energy ${E}_{0}^{\prime}$ differs from its value E _{0} in the empty Minkowski spacetime. The difference ${E}_{0}^{\prime}{E}_{0}$ can be measured and hence it has a well‐defined physical meaning. This phenomenon is known as the Casimir effect.
(p.445) Denote by n _{k} a non‐negative integer number for a mode k and by {n _{k}} a set of such integer numbers. We assume that only a finite number of them do not vanish. In the occupation number representation the basis in the Hilbert space of states is
This basis is orthonormal
F.3.3 Thermal fields
A thermal state of the quantum field with the inverse temperature β is a state when each of its modes is in a thermal state described by the density matrix Eq. (F.2.14). The corresponding density matrix for the quantum field is
Here, Ĥ is a complete Hamiltonian Eq. (F.3.17) for the quantum field and Ĥ_{0} is its version with zero‐point energy subtracted. The trace operator is the trace over the complete Hilbert space of the field states. Using the representation of the field as a set of decoupled oscillators, one obtains
This is a free energy of the thermal quantum field at the temperature Θ = β ^{−1}.
F.3.4 Continuous spectrum
The imposed boundary conditions, that is vanishing of the field at the boundary, are somehow artificial. However, if the quantization box is large and one is studying observables in a domain far away from the boundary, the result does not depend on the particular boundary conditions. They affect the state of the field only close to the boundary. Usually, for large volume the surface effects can be neglected.
In order to exclude effects connected with the spatial location of the boundary it is convenient to use the other boundary conditions, known as the periodicity conditions. Let us again take a cube with boundaries at ǀX _{i}ǀ = L _{i}/2 and impose the following periodic boundary conditions
In other words, the opposite boundaries of the cube are identified with one another and we have a space that is 3‐torus 𝕋^{3}.
(p.446) The complete set of eigenfunctions of the Laplace operator with given periodicity conditions is well known. For each of three orthogonal directions one can use sin(k _{i} X _{i}) and cos(k _{i} X _{i}) functions. The periodicity conditions imply that
where n _{i} are non‐negative integer numbers. The ‘doubling’ of the basic functions is ‘compensated’ by the fact that the corresponding wavelengths k _{i} are twice ‘rare’ than before. Very often it is more convenient instead of real eigenfunctions of the Laplace operator to use the complex eigenfunctions exp(±ik _{i} X _{i}). These functions also obey the periodicity conditions when Eq. (F.3.24) is satisfied. We use the following solutions
Here, $k=\left(\frac{2\pi {n}_{1}}{{L}_{1}},\frac{2\pi {n}_{2}}{{L}_{2}},\frac{2\pi {n}_{3}}{{L}_{3}}\right)$ where n _{i} are the integer numbers, which can be both positive and negative. Under this condition the real basic functions sin(k _{i} X _{i}) and cos(k _{i} X _{i}) can be obtained as linear combinations of solutions Φ _{k} and their complex conjugated. The basis functions obey the following normalization conditions
A mode decomposition of the field operator ϕ̂(t, X) now takes the form
Making the size of the quantization box infinitely large one reduces the distance between the nearby wavelength, so that finally in the infinite‐size limit one obtains a continuous spectrum of the Laplace operator.
The continuous spectrum formulas can be obtained from the discrete ones by the following substitution
$$\begin{array}{l}{\Phi}_{k}\to A\text{}{\Phi}_{k}\text{,}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}{\widehat{a}}_{k}\to A\text{}{\widehat{a}}_{k}\text{,}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}{\widehat{a}}_{k}^{\u2020}\to A{\widehat{a}}_{k}^{\u2020},\hfill \\ {\displaystyle \sum _{k}}\to {A}^{2}{\displaystyle \int d}k\text{,}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}{\delta}_{k,{k}^{\prime}}\to {A}^{2}\text{}\delta (k{k}^{\prime}),\hfill \\ \left[{\widehat{a}}_{k},{\widehat{a}}_{{k}^{\prime}}^{\u2020}\right]={\delta}_{k,{k}^{\prime}}\to \left[{\widehat{a}}_{k},{\widehat{a}}_{{k}^{\prime}}^{\u2020}\right]=\delta (k{k}^{\prime}),\hfill \end{array}$$(F.3.28)where $A=\sqrt{V/{(2\pi )}^{3}}$ The discrete and continuous δ‐functions are defined as usual to fulfill the relations ∑_{k} δ _{k,k′}= 1 and ∫ d k δ(k ‐ k′) = 1.
The continuous spectrum basis functions are
(p.447) They obey the normalization condition
The field representation in the limit of an infinite box is
where the operators of creation and annihilation with continuous index k obey the commutation relations
F.3.5 Green functions
A Green function for a quantum field φ̂ is an average value of a bilinear combination of φ̂(t,X) and φ̂(t′,X′). Again, as in the case of a quantum oscillator, there exists a variety of Green functions that differ by the choice of a state used for averaging and by a concrete form of the bilinear form, which may include special operator ordering.
Let us consider the following thermal correlators
where
These functions are known as thermal Wightman functions. Since the spacetime is static one has
This property can be proved by using the relations
and the fact that cyclic permutations of operators within the Tr ‐operation do not change the result.
The property Eq. (F.3.35) means that ${G}_{\beta}^{+}$ and ${G}_{\beta}^{}$ functions depend only on the difference t − t′. Moreover, since the system is invariant under spatial translations and rotations, these fuctions depend only on r = ǀX − X′ǀ. Thus, we have
(p.448) To calculate these quantities it is sufficient to substitute Eq. (F.3.31) in their definition Eq. (F.3.33), and to use the relations
The calculations give
Here, ${\mathcal{G}}_{\beta ,\omega}^{\pm}(t)$ is the corresponding Green function for a thermal oscillator with the frequency co given by Eq. (F.2.29). Similar representations are valid for the other Green functions of the quantum field.
Let G, G _{f,β}, and ${\mathcal{G}}_{\beta}^{(1)}$ be the thermal average of the commutator, the Feynman propagator and the Hadamard function for the quantum field (Takagi 1986; Birrell and Davies 1982), respectively
Here,
To obtain any of these functions it is sufficient to substitute in the relation similar to Eq. (F.3.39) a corresponding Green function for the oscillator
Here, the • ‐symbol specifies a concrete choice of the Green function.
The thermal oscillatory Green functions consists of two parts. The first, which coincides with the vacuum average, differs for different Green functions, while the second (which vanishes at the zero‐temperature limit and may be absent in some cases) is a universal expression
In a special case of a massless field, m = 0, the expressions for the Green functions can be written in terms of the elementary functions. One has ω = k, and the integral in Eq. (F.3.42) can be easily calculated. For B _{β} (t, r) related to ℬ(t) by Eq. (F.3.42) one finds
Using this result and the following relation
where ϵ 〉 0 is an infinitesimal quantity and Ƥ is the principal value, one obtains the real‐time thermal Green functions for massless fields
For imaginary time t = −iτ the Wightman functions are equal, ${G}_{\beta}^{+}(i\tau ,r)={G}_{\beta}^{}(i\tau ,r)$ and the Matsubara propagator is
It is periodic in the imaginary time with the period β
and obeys the following equation
For r = 0
Thermal Green functions for massive fields and fields of other spins can be obtained in a similar manner. In all cases the general property remains valid: The quantum field at the finite (p.450) temperature α ^{1} is connected by analytical continuation to the Euclidean field theory on the Euclidean space with the topology of the cylinder S ^{1} × ℝ^{3}, where the length of S ^{1} is β.
F.3.6 Measurement of temperature
Let us discuss now how the temperature of the radiation can be measured. In a general setup, a thermometer is a system with internal degrees of freedom that can interact with thermal radiation. We assume that this interaction is weak and does not disturb the state of the radiation. We expect that the internal degrees of freedom of the thermometer will be excited as a result of an interaction and, after some time, they will be distributed thermally. By ‘measuring’ observables depending on this distribution one can ‘measure’ temperature of the radiation. Usually, it is assumed that the thermometer has small size. This is important when the thermal radiation is affected by an external field. For example, in the presence of a static gravitational field the local temperature depends on a point. In order to measure this local temperature the size of the thermometer must be much smaller than a characteristic scale at which the temperature changes.
For illustration let us consider a simple model. We choose a quantum oscillator as a thermometer and consider its interaction with a scalar massless field ϕ. Let the ‘thermometer’ be located at a point X _{0} of a flat spacetime. The system, that is the quantum field ϕ and the oscillator q, is described by the following action
Here, S _{0}[ϕ] is the field action Eq. (F.3.1), S _{0}[q] is the oscillator action Eq. (F.1.1). The interaction action is
where ϕ(t) = ϕ(t, X _{0}). We denote by Ĥ the Hamiltonian of the complete system. It has the form
In the Schrodinger picture the evolution of the complete system is described by the state vector ǀΨ(t)} obeying the Schrödinger equation
In this picture the operators do not evolve, so that
Let us write
(p.451) Substitution of Eq. (F.3.56) into Eq. (F.3.54) gives
Denote by ǀA⟩ the vectors of the orthonormal basis in the Hilbert space of the states of the system without interaction, and A is an index enumerating the vectors of this set. We additionally assume that ǀA⟩ are eigenstates of the free Hamiltonian
Let us write
Substituting this relation into Eq. (F.3.57) and multiplying the obtained relation by ⟨Bǀ we get
We assume that the interaction constant λ is small and use the perturbation theory to solve the Eq. (F.3.60). In the first‐order approximation, the amplitude of the probability of the transition from the initial state ǀA⟩ to the final state ǀB⟩ at time t is
The probability of the transition A → B is
Till now the consideration was quite general and we did not use specific properties of the interaction. Basically, the relation Eq. (F.3.62) reproduces the standard quantum mechanical result of the perturbation theory for the probability of transitions under a general perturbation V̂. In order to apply this result to our system of a quantum oscillator interacting with the thermal radiation we proceed as follows. First, note that a state ǀA⟩ of the complete system can be written as
where ǀn⟩ is a complete set of the states of a noninteracting quantum oscillator and ǀN⟩ is a complete set of the field states. Accordingly, we write the index A = {n, N}. For the index B we write B = {m,M}. We choose the states ǀn) and ǀN⟩ to be the eigenstates of the corresponding free Hamiltonian
(p.452) In this basis one has
We used here that $\u3008m\widehat{q}(t)n\u3009={e}^{i({\omega}_{m}{\omega}_{n})t}{q}_{mn}$. These matrix elements can be found by using the expression Eq. (F.2.7)
where ω is the frequency of the oscillator. One has
The relations Eq. (F.3.65) and Eq. (F.3.67) show that in the lowest order of the perturbation theory the transitions are only of two types:

• either the oscillator absorbs a single ‘photon’ of energy ω and ‘jumps’ from the lower level n to the upper level n + 1;

• or it emits a ‘photon’ of the frequency ω; and ‘jumps’ from the upper level n + 1 to the lower level n.
To obtain the probability for the transition w _{mn} one needs to average over unobserved final states of the field ǀM⟩ and to use the thermal density matrix for the initial state of the field
Here, P _{N} is a probability of a state ǀN⟩. As a result of these operations one gets
Here,
is the positive‐frequency Wightman function
It is easy to show that a double integral in Eq. (F.3.69) can be rewritten as follows
(p.453) This relation shows that w _{mn} is proportional to time. Thus, for the transision probability per unit time one has
where
Let us denote ẇ_{n+} = ẇ_{n+1,n} and ẇ_{n−} = ẇ_{n,n+1}. Equation (F.3.67) shows that
Thus,
Simple calculations give
for either positive or negative ω. Thus,
The obtained relation implies that the ratio of the probability of the transition n → n + 1 to the probability of the inverse process n + 1 → n is a universal function. It does not depend on the details of interaction and it is determined only by the temperature of the radiation β ^{−1} and the energy difference of the corresponding levels. This ratio is determined only by the energy spectrum of the Wightman function G̃(ω) for the radiation. This result is quite general and it allows one to prove that the equilibrium state of the quantum oscillator interacting with the thermal bath with temperature β ^{−1} is described by the thermal density matrix with the same temperature.
This can be easily shown as follows. Denote by p _{n}(t) the probability of the oscillator to be at the level n. Then, the change in time of this probability is
For n = 0 one has
In the equilibrium state ṗ_{n} = 0. The relation Eq. (F.3.80) gives
(p.454) Starting with this relation and using Eq. (F.3.79) we obtain
The normalization condition ${\sum}_{n=0}^{\infty}{p}_{n}}=1$determines p _{0}
Thus, the equilibrium distribution of the oscillator over the energy levels has a thermal law. The details of its interaction with the thermal bath, and especially the value of the coupling constant λ determine the rate of transitions ẇ_{n±}, and, in particular, how fast the equilibrium is reached. The smaller λ the longer one needs to wait when the oscillator reaches its equilibrium thermal state. In other words, if the interaction is weak the time required to ‘measure’ the temperature of the thermal radiation is large.
F.4 Quantum Theory in (1+1)‐Spacetime
F.4.1 Field equations
Till now we have considered the case when a quantum field is freely propagating in a flat spacetime, so that the mode functions are the standard plane waves. Let us consider a slightly more complicated situation when the scalar field moves in a space with a potential. For simplicity we assume that the space has only one dimension and denote a spatial coordinate by x. The action for the field ϕ(t, x) is
Here, U(x) is the potential of the form shown in Figure F.2. It is positive and falls rapidly enough at ǀxǀ → ∞. One can also consider this action as a generalization of the (1 + 1)‐ dimensional version of Eq. (F.3.1) to the case when the mass of the field depends on x.
The field equation is
(p.455) The field can be decomposed into modes
where u _{k} are complex solutions of the following eigenvalue problem
The latter equation is simply a standard Schrödinger equation of quantum mechanics
It describes a one‐dimensional motion of a particle of the energy E = k ^{2} in the presence of the potential barrier. This problem is discussed in standard books on quantum mechanics. Here, we just collect the main results that are required for our analysis.
F.4.2 Bases in the solution space
The spectrum of the problem Eq. (F.4.5) is continuous and the energy E is positive. The energy levels are degenerate, namely there exist two linear independent solutions for a given energy E. It is convenient to consider complex solutions of Eq. (F.4.5). The scalar product for any two complex solutions f _{1} and f _{2} is defined as follows
A complex solution is uniquely specified by its asymptotic behavior at x = ±∞. In the general case, for a given k 〉 0 one has
For the continious spectrum the scalar product can be written in terms of the scattering coefficients α _{k}, β _{k},γ _{k} and δ _{k}. Since a solution of the original wave equation Eq. (F.4.2) is obtained by multiplying f _{k}(x) by e ^{−ikt}, it is evident that a term with e ^{−ikx} describes a mode propagating to the left, while a mode with e ^{ikx} moves to the right. This can be presented by the following conformal diagram (see Figure F.3).
One can interpret the solution Eq. (F.4.7) as follows. An initial wave consists of two parts: One, with the amplitude β _{k,} is going from x = −∞ to the right, and the other, with the amplitude α _{k,} is propagating from x = 0∞ to the left. After the scattering on the potential U(x) the right‐moving mode will have the amplitude δ _{k} at x = ∞, while the left‐moving mode will have the amplitude γ _{k} at x = −∞. Since the mode is uniquely specified by its initial data α _{k} and β _{k}, the ‘future’ asymptotic data, γ _{k} and δ _{k}, are functions of the initial data. These functions depend on the form of the potential U(x). For any two solutions Eq. (F.4.7) one has
We shall use the normalization condition
A standard convenient choice of the complete set of normalized complex solutions consists of functions u _{in} and u _{up}. These functions are specified by the following scattering data
In these solutions we assume that k 〉 0. The conformal diagrams for these functions are shown in Figure F.4.
The normalization condition Eq. (F.4.9) results in
These solutions obey the following relations
It is well known (and easy to demonstrate) that the Wronskian
of any two linearly independent solutions ν _{1} and ν _{2} to Eq. (F.4.5) is a constant. By calculating the Wronskian at both infinities, x = ±∞, for solutions Eqs. (F.4.10) and (F.4.11) and their (p.457)
A solution u _{up,k} describes a stationary wave propagating from x = −∞. It is partially scattered back and partially passes through the potential barrier. The coefficients r _{k} and t _{k} are the reflection and transmission coefficients, respectively. Similarly, u _{in,k}(x) describes a stationary wave propagating from x = ∞, and R _{k} and T _{k} are its reflection and transmission coefficients. Equation (F.4.15) shows that the transmission amplitudes T _{k} and t _{k} are the same, while for the reflection amplitudes one has ǀR _{k}ǀ = ǀr _{k}ǀ, while their phases can be different.
Since the potential in Eq. (F.4.4) is real, one can always choose basic solutions to be real. For each of k 〉 0 one has two real solutions that we denote ν _{1,k} and ν _{2,k}. The conformal diagram for these solutions are shown in Figure F.5. These solutions are special versions of a general solution Eq. (F.4.7). For ${v}_{1,k}{\alpha}_{k}={\delta}_{k}^{*}={C}_{+,k}$and ${\gamma}_{k}={\beta}_{k}^{*}={C}_{,k}$. Similarly, for ${v}_{2,k}{\alpha}_{k}={\delta}_{k}^{*}={D}_{+,k}$and ${\gamma}_{k}={\beta}_{k}^{*}={D}_{,k}$The normalization and orthogonality conditions for ν _{1,k} and ν _{2,k} give
The coefficients C _{+,k} and D _{+,k} are uniquely determined by the coefficient C _{−,k} and D _{−k}. The corresponding relations include the reflection and transmission coefficient. One can use the following asymptotic data, specifying the real solutions ν _{1,k} and ν _{2,k}:
Here,${z}_{k}=\sqrt{1+\left{t}_{k}\right}$
Basic solutions u _{in,k} and u _{up,k} can be written as linear combinations of ν _{1,k} and ν _{2,k}
where
Denote
It is evident that this function is real and symmetric with respect to its arguments x and x′
Using Eqs. (F.4.18) and (F.4.19) one can check that
At far distances, where the potential U(x) becomes small, the function V _{k} has the following asymptotic form
In this expression … denote omitted terms that either decrease at infinity or contain the fast‐ oscillating factor ∼exp[±ik(x + x′)]. The latter does not contribute to the integral over k.
(p.459) F.4.3 Quantization
To quantize the scalar field ϕ one can write it in the following two equivalent forms
In the first line, the operators q̂_{1,k}(t) and q̂_{2,k}(t) are standard oscillatory‐amplitude operators. In the second and third lines, ${\widehat{a}}_{k}^{\u2020},{\widehat{b}}_{k}^{\u2020},{\widehat{a}}_{k}$ and b̂_{k} are the operators of creation and annihilation obeying the relations
All other independent commutators vanish. The operator ${\widehat{a}}_{k}^{\u2020}$ creates left‐moving quanta in the mode u _{in,k,} while ${\widehat{b}}_{k}^{\u2020}$ creates right‐moving quanta in the mode u _{up,k}. The vacuum state is a state when neither left‐moving nor right‐moving quanta are present. It is defined by the conditions
The Hamiltonian of the system (with zero‐point fluctuation contribution excluded) is
F.4.4 Equilibrium thermal state
Thermal equilibrium state with temperature Θ = β ^{−1} is described by the thermal density matrix
The thermal Wightman functions are defined as
Using Eq. (F.4.24) one obtains
(p.460) Similar expressions are valid for all other Green function
Here, the •‐symbol specifies the corresponding Green function.
In particular, the Matsubara propagator is
This propagator is periodic in τ with the period β and obeys an equation
The Matsubara propagator can be obtained as a Green function on the Euclidean cylinder S ^{1} ×ℝ^{1} in the presence of the potential U and with the size of the compact dimension equal to β (see Figure F.6).
F.4.5 Local observables
As an application of the developed formalism, let us calculate the asymptotic value of the thermal averages of the stress‐energy tensor for the massless field ϕ̂. To obtain the temperaturedependent part of T _{μν} we shall use the renormalized Hadamard function
In the asymptotic region, where the potential U(x) vanishes, the stress‐energy tensor can be written as follows
(p.461) Simple calculations give
Thus, outside the potential for the thermal state one has p = ϵ. This result is in agreement with the expected one.
F.4.6 Non‐equilibrium thermal state
The density matrix Eq. (F.4.28) for the thermal equilibrium state can be written as a product of two independent density matrices
Here, Ĥ_{in} and Ĥ_{up} are the Hamiltonians for in‐ and up‐quanta given by Eq. (F.4.27). In the thermal equilibrium state the in‐ and up‐quanta are emitted with the same temperature β ^{−1} from x = oo and x = − ∞, respectively. Since Ĥ_{in} and H ̂_{up} are independent, one can consider a more general state, which is described by the following density matrix
For β _{in} ≠ β _{up} this state is stationary, but not inequilibrium. In such a state there exists a net flux of thermal radiation from the hotter source to the colder one. We call such a state a non‐equilibrium thermal state.
The renormalized Hadamard function for this state is
and D _{up} is obtained from D _{in} by changing in → up. In these relations {c.c.} means a complex conjugated expression, and
To calculate the renormalized stress‐energy tensor at x →∞ we use the following asymptotics^{3}
Here,… denote the terms that either vanish at infinity or rapidly oscillate there, ~ exp[ik(x + x′)]. These terms will not contribute to the asymptotic value of the T _{μν}. Denote ν = t + x and u = t − x, then, (p.462)