In–Out, In–In, and Schwinger–Keldysh Functionals

The previous page described vacuum decay through the vacuum persistence amplitude. That already forced a distinction that is easy to miss in ordinary perturbation theory. A quantity such as

$$\langle 0_{\rm out}|T\{\phi(x)\phi(y)\}|0_{\rm in}\rangle$$

is a transition matrix element. It is the right object for scattering amplitudes, the ordinary Feynman effective action, and vacuum-to-vacuum transitions. It is not, by itself, the expectation value measured by an observer who prepares a state in the past and asks what an operator is at time $t$.

This page develops the real-time formalism that keeps that distinction honest. The ordinary Feynman, or in–out, formalism fixes both an initial and a final vacuum. The in–in, or Schwinger–Keldysh, formalism fixes only the initial state and evolves both the ket and the bra. The reward is causality: the response of an operator at time $t$ depends only on the past of $t$, even though Feynman propagators themselves are not retarded functions.

The conceptual slogan is short:

$$\text{in–out computes transition amplitudes,} \qquad \text{in–in computes expectation values.}$$

Required background. Vacuum persistence and its imaginary part explains why a vacuum-to-vacuum amplitude need not have unit modulus, while the Feynman propagator and its $i\epsilon$ prescription supplies the in–out boundary condition used below.

Helpful background. Hyperbolic equations and causal propagators reviews causal support and retarded or advanced Green operators.

Formalism	Boundary data	What it naturally computes	Typical Green function
in–out	initial vacuum and final vacuum	scattering amplitudes, vacuum persistence, ordinary effective action	Feynman propagator
in–in / Schwinger–Keldysh	initial density matrix only	expectation values, currents, particle number, backreaction	retarded and Wightman functions

The table is a good diagnostic while reading the rest of this page: whenever the desired answer is a number measured at time $t$ in a prepared state, the in–in column is the safer starting point.

Time evolution and vacuum persistence

Correlator convention

This lesson follows the manuscript and defines contour correlators without a leading $-i$:

$$D_{\sigma\sigma'}=\langle T_C\phi_\sigma\phi_{\sigma'}\rangle, \qquad D_R(x,y)=\theta(x^0-y^0)\langle[\phi(x),\phi(y)]\rangle.$$

Many-body texts instead write $G^{\sigma\sigma'}=-iD_{\sigma\sigma'}$ and $G^R=-iD_R$. For a probe $H\to H-hB$, the convention here gives $\delta\langle A\rangle/\delta h=iD_R=-G^R$. Keep this overall factor and the probe sign fixed when translating formulas.

Work in the interaction picture and let

$$S(t_2,t_1)=T\exp\left[-i\int_{t_1}^{t_2}dt\,H_I(t)\right]$$

for $t_2>t_1$. The initial state is prepared at $t=-\infty$ and is denoted $|0\rangle_{\rm in}$. The state obtained by evolving it to the future is

$$S(\infty,-\infty)|0\rangle_{\rm in}.$$

Its overlap with the future vacuum is the vacuum persistence amplitude,

$$S_{00} ={}_{\rm out}\langle0|0\rangle_{\rm in} =\langle0|S(\infty,-\infty)|0\rangle.$$

A convenient notation is

$$S_{00}=e^{iW[0]}.$$

The functional $W[0]$ is the sum of connected vacuum bubbles. In ordinary perturbation theory the denominator in normalized Feynman correlators cancels disconnected vacuum bubbles:

$$\langle T\phi(x_1)\cdots\phi(x_n)\rangle_{\rm F} = {\langle0|T\{\phi(x_1)\cdots\phi(x_n)S(\infty,-\infty)\}|0\rangle \over \langle0|S(\infty,-\infty)|0\rangle}.$$

When the vacuum is stable, $S_{00}$ is a phase. When the vacuum can decay, or when an external field can create particles, $|S_{00}|<1$. Writing

$$W[0]=\operatorname{Re}W[0]+i\operatorname{Im}W[0],$$

one finds

$$|S_{00}|^2=e^{-2\operatorname{Im}W[0]}.$$

Thus an imaginary part of the in–out effective action is not an inconsistency. It says that the vacuum-to-vacuum channel is not closed. Probability has gone into other final states.

The in–out object evolves from the past to the future and projects onto a final vacuum. The in–in object evolves a ket forward to the operator and evolves the bra back along the conjugate path, so it is an expectation value in the state prepared in the past.

In–out matrix elements are not expectation values

Let $B(t)$ be a Heisenberg operator, represented in the interaction picture by inserting the appropriate evolution operators. The in–out quantity associated with the ordinary Feynman functional is

$$\boxed{ \langle B(t)\rangle_{\rm in\text{-}out} = {\langle0|S(\infty,t)B_I(t)S(t,-\infty)|0\rangle \over \langle0|S(\infty,-\infty)|0\rangle}. }$$

The factor $S(\infty,t)$ is the giveaway. This expression knows about the future boundary condition. It is a transition matrix element between an initial vacuum and a final vacuum, divided by their overlap.

The actual expectation value in the state prepared at $t=-\infty$ is instead

$$\boxed{ \langle B(t)\rangle_{\rm in\text{-}in} = \langle0|S^\dagger(t,-\infty)B_I(t)S(t,-\infty)|0\rangle. }$$

No evolution beyond $t$ is needed. The ket evolves forward to $t$; the bra evolves backward from $t$ to the same initial density matrix. In a pure state this is simply the ordinary expectation value

$$\langle B(t)\rangle =\langle\Psi(t)|B_I(t)|\Psi(t)\rangle, \qquad |\Psi(t)\rangle=S(t,-\infty)|0\rangle.$$

The distinction is invisible in some equilibrium calculations because the same Feynman diagrams are often repurposed after analytic continuation or after extracting on-shell amplitudes. It becomes essential for time-dependent backgrounds, vacuum decay, cosmology, transport, particle production, and any problem where the question is “what does the system do?” rather than “what is the transition amplitude between two vacua?”

The closed time path

The Schwinger–Keldysh construction encodes the in–in expression as a path integral on a doubled time contour. Prepare a density matrix $\rho_0$ at time $t_0$, choose a return time $t_f$ later than every operator insertion, evolve forward on a $+$ branch, and evolve backward on a $-$ branch. The return time is bookkeeping, not a future boundary condition; it may be moved without changing observables as long as it remains later than all insertions. Fields on the two branches are denoted

$$\phi_+(x), \qquad \phi_-(x).$$

Sources are also doubled. The closed-time-path generating functional is

$$\boxed{ Z_{\rm SK}[J_+,J_-] =\operatorname{Tr}\left[ U_{J_+}(t_f,t_0)\rho_0 U_{J_-}^\dagger(t_f,t_0) \right]. }$$

For a pure initial state $\rho_0=|\Psi_0\rangle\langle\Psi_0|$, cyclicity of the trace gives

$$Z_{\rm SK}[J_+,J_-] = \langle\Psi_0|U_{J_-}^\dagger(t_f,t_0) U_{J_+}(t_f,t_0)|\Psi_0\rangle.$$

In path-integral form,

$$Z_{\rm SK}[J_+,J_-] = \int_{\phi_+(t_f)=\phi_-(t_f)} \mathcal D\phi_+\mathcal D\phi_-\, \rho_0[\phi_+(t_0),\phi_-(t_0)] \exp\left\{ iS_{t_0}^{t_f}[\phi_+]-iS_{t_0}^{t_f}[\phi_-] +i\int_{t_0}^{t_f}d^dx\,(J_+\phi_+-J_-\phi_-) \right\},$$

The two fields are sewn together and integrated over at $t_f$. The minus sign in front of $S[\phi_-]$ is not optional: it is the complex-conjugate time evolution of the bra. A non-Gaussian $\rho_0$ also produces interaction vertices localized at the initial surface.

The Schwinger–Keldysh contour doubles the fields. The $+$ branch evolves forward with $e^{iS}$, while the $-$ branch evolves backward with $e^{-iS}$. The two histories are sewn at any $t_f$ later than the latest insertion; equal sources then give $U^\dagger U=1$.

The normalization property is crucial:

$$Z_{\rm SK}[J,J] =\operatorname{Tr}\left[U_J(t_f,t_0)\rho_0U_J^\dagger(t_f,t_0)\right] =\operatorname{Tr}\rho_0 =1.$$

This identity assumes a normalized density matrix and a Hermitian source coupling, so that evolution with the common source is unitary. Closed-time-path vacuum bubbles then cancel against their conjugates. In the in–out formalism, vacuum bubbles exponentiate into $S_{00}=e^{iW}$. In the in–in formalism with equal sources, the corresponding closed contour gives unity. That is the diagrammatic reason the Schwinger–Keldysh formalism is naturally normalized for expectation values.

Four real-time propagators

Contour ordering produces four two-point functions. Define

$$D_{\sigma\sigma'}(x,y) =\langle T_C\phi_\sigma(x)\phi_{\sigma'}(y)\rangle, \qquad \sigma,\sigma'\in\{+,-\}.$$

For a translation-invariant vacuum initial state in a free scalar theory,

$$D_{++}(x,y)=\langle T\phi(x)\phi(y)\rangle,$$ $$D_{--}(x,y)=\langle \widetilde T\phi(x)\phi(y)\rangle,$$ $$D_{-+}(x,y)=\langle\phi(x)\phi(y)\rangle, \qquad D_{+-}(x,y)=\langle\phi(y)\phi(x)\rangle.$$

Here $T$ is time ordering and $\widetilde T$ is anti-time ordering. With the Fourier convention of the course,

$$D_{++}(p)={i\over p^2-m^2+i\epsilon},$$ $$D_{--}(p)={-i\over p^2-m^2-i\epsilon},$$ $$D_{-+}(p)=2\pi\theta(p^0)\delta(p^2-m^2), \qquad D_{+-}(p)=2\pi\theta(-p^0)\delta(p^2-m^2).$$

The mixed propagators are on shell. They are the real-time Wightman functions, and in cut diagrams they are the lines that cross from an amplitude to its complex conjugate. For a thermal or populated Gaussian state the same free spectral support remains, but the positive- and negative-frequency terms acquire occupation factors. For an interacting or nonstationary state, the compact vacuum momentum-space formulas above no longer apply even though the contour definitions do.

The doubled contour gives a $2\times2$ matrix of real-time propagators. The diagonal entries are Feynman and anti-Feynman propagators; the off-diagonal entries are Wightman functions, supported on the mass shell in momentum space.

These four functions are not independent. Since

$$D_{++}(x,y)=\theta(x^0-y^0)D_{-+}(x,y)+\theta(y^0-x^0)D_{+-}(x,y),$$

and

$$D_{--}(x,y)=\theta(y^0-x^0)D_{-+}(x,y)+\theta(x^0-y^0)D_{+-}(x,y),$$

one has, with a common prescription at coincident times,

$$\boxed{ D_{++}+D_{--}=D_{+-}+D_{-+}. }$$

The retarded function is the difference

$$D_R(x,y) =\theta(x^0-y^0)\langle[\phi(x),\phi(y)]\rangle =D_{++}(x,y)-D_{+-}(x,y) =D_{-+}(x,y)-D_{--}(x,y).$$

The advanced partner is

$$D_A(x,y) =-\theta(y^0-x^0)\langle[\phi(x),\phi(y)]\rangle =D_{++}(x,y)-D_{-+}(x,y) =D_{+-}(x,y)-D_{--}(x,y).$$

At equal times, contact terms and the convention for $\theta(0)$ must be treated consistently. Away from coincident points, these identities are purely algebraic. They show how causal response is built from Feynman and Wightman pieces so that the nonretarded parts cancel.

Branch signs and cut diagrams

Perturbation theory on the closed contour has a simple rule. A vertex on the $+$ branch carries the ordinary factor from $iS[\phi_+]$. A vertex on the $-$ branch carries the complex-conjugate factor from $-iS[\phi_-]$. For example, if

$$S_{\rm int}[\phi]=\int d^dx\,\left(-{\lambda\over4!}\phi^4\right),$$

then a $+$ vertex contributes

$$-i\lambda,$$

whereas a $-$ vertex contributes

$$+i\lambda.$$

Internal lines know which branches their endpoints live on:

$$+\,+\quad\text{uses }D_{++}, \qquad -\,-\quad\text{uses }D_{--}, \qquad -\,+\text{ or }+\,-\quad\text{uses a Wightman function.}$$

The diagrammatic structure resembles the decomposition of a probability into an amplitude and a conjugate amplitude. The left side of a cut contains ordinary Feynman propagators, the right side contains complex-conjugate propagators, and the cut lines are on shell. Schematically,

$${1\over p^2-m^2+i0} \quad\text{on the amplitude side},$$ $${1\over p^2-m^2-i0} \quad\text{on the conjugate side},$$

and

$$\theta(p^0)\delta(p^2-m^2) \quad\text{across the cut}.$$

Up to conventional factors of $i$ and $2\pi$, these are exactly the ingredients displayed by the Schwinger–Keldysh matrix. This is why closed-time-path perturbation theory is the natural language for inclusive probabilities, particle production, transport, and nonequilibrium evolution.

The Keldysh rotation

The doubled fields are often reorganized into “classical” and “quantum” combinations,

$$\phi_c={\phi_++\phi_-\over2}, \qquad \phi_q=\phi_+-\phi_-.$$

Similarly define

$$J_c={J_++J_-\over2}, \qquad J_q=J_+-J_-.$$

The source term becomes

$$J_+\phi_+-J_-\phi_-=J_c\phi_q+J_q\phi_c.$$

The physical limit of equal sources is

$$J_+=J_-=J, \qquad J_c=J, \qquad J_q=0.$$

Define the connected functional by

$$W_{\rm SK}[J_c,J_q]=-i\log Z_{\rm SK}[J_c,J_q].$$

This notation makes unitarity visible. The action difference

$$S[\phi_+]-S[\phi_-]$$

vanishes when $\phi_q=0$, and the generating functional satisfies $Z_{\rm SK}[J_c,J_q=0]=1$. An ordinary expectation value is obtained from the auxiliary difference source:

$$\langle\phi_c(x)\rangle_J =\left.{\delta W_{\rm SK}\over\delta J_q(x)}\right|_{J_q=0}.$$

The response to the physical common source $J_c$ is

$${\delta\langle\phi_c(x)\rangle_J\over\delta J_c(y)} =\left.{\delta^2W_{\rm SK}\over \delta J_c(y)\,\delta J_q(x)}\right|_{J_q=0} =iD_R(x,y),$$

for the source and correlator conventions of this page. This makes the index logic precise: the observable comes from a $J_q$ derivative, while varying the physical source adds a $J_c$ derivative.

The same rotation organizes the two-point functions into

$$\begin{pmatrix} \langle\phi_c(x)\phi_c(y)\rangle & \langle\phi_c(x)\phi_q(y)\rangle\\ \langle\phi_q(x)\phi_c(y)\rangle & \langle\phi_q(x)\phi_q(y)\rangle \end{pmatrix} = \begin{pmatrix} F(x,y) & D_R(x,y)\\ D_A(x,y) & 0 \end{pmatrix},$$

where

$$F(x,y)={1\over2}\langle\{\phi(x),\phi(y)\}\rangle$$

is the statistical correlator. The zero in the $qq$ entry is another expression of the closed-contour normalization. Dynamics and state information are thereby separated: $D_R$ and $D_A$ carry causal propagation, while $F$ carries occupation and fluctuations.

Causal response from the closed contour

The simplest test is first-order perturbation theory. Let $B(t)$ be a local operator and let

$$H_I(t)=\int d^{d-1}\mathbf x\,\mathcal H_I(t,\mathbf x).$$

The in–in expectation value is

$$\langle B(t)\rangle =\langle0|S^\dagger(t,-\infty)B_I(t)S(t,-\infty)|0\rangle.$$

Expanding to first order gives

$$S(t,-\infty)=1-i\int_{-\infty}^{t}dt'\,H_I(t')+O(H_I^2),$$ $$S^\dagger(t,-\infty)=1+i\int_{-\infty}^{t}dt'\,H_I(t')+O(H_I^2).$$

Therefore

$$\boxed{ \delta\langle B(t)\rangle =i\int_{-\infty}^{t}dt'\, \langle[H_I(t'),B_I(t)]\rangle_0. }$$

This is a retarded formula. The upper limit is $t$, and the commutator makes microscopic causality manifest. If $B(t)$ and $\mathcal H_I(x)$ are spacelike separated local operators, then their commutator vanishes.

For example, take

$$B(0)=\phi^2(0), \qquad H_I(t)=\int d^{d-1}\mathbf x\,{\lambda\over4!}\phi^4(t,\mathbf x).$$

Then

$$\delta\langle\phi^2(0)\rangle =i{\lambda\over4!}\int_{-\infty}^{0}d^dx\, \langle[\phi^4(x),\phi^2(0)]\rangle_0.$$

Only points $x$ in the past light cone of the origin can contribute. The $+$ and $-$ branches have combined to turn a time-ordered object into a commutator.

The first-order in–in response of a local operator at the origin is an integral over a commutator. By microcausality, the integrand vanishes outside the past light cone. Future insertions cancel between the forward and backward branches.

The corresponding in–out correction is instead

$$\delta\langle B(t)\rangle_{\rm in\text{-}out} =-i\int_{-\infty}^{\infty}dt'\, \langle T\{B_I(t)H_I(t')\}\rangle_{0,\rm conn}.$$

This is the correct first variation of an in–out transition amplitude. It is not a causal expectation value, because it is not trying to be one.

Oscillator Green functions and state dependence

The same lesson can be seen without fields. To avoid using one symbol for two different quantities, let $M$ be the mechanical mass and $\Omega$ the oscillator frequency:

$$q(t)={1\over\sqrt{2M\Omega}} \left(ae^{-i\Omega t}+a^\dagger e^{i\Omega t}\right), \qquad [a,a^\dagger]=1.$$

The ground-state Feynman function is

$$D_F(t,t')=\langle0|Tq(t)q(t')|0\rangle ={1\over2M\Omega}e^{-i\Omega|t-t'|}.$$

The cusp at $t=t'$ supplies the delta function:

$$M(\partial_t^2+\Omega^2)D_F(t,t')=-i\delta(t-t').$$

Fourier transforming gives

$$M(\omega^2-\Omega^2)D_F(\omega)=i,$$

and the ground-state boundary condition selects

$$D_F(\omega)={i\over M(\omega^2-\Omega^2+i0)}.$$

The differential equation alone does not fix the Green function. Because

$$(\omega^2-\Omega^2)\delta(\omega-\Omega)=0, \qquad (\omega^2-\Omega^2)\delta(\omega+\Omega)=0,$$

one may add homogeneous on-shell pieces:

$$D(\omega)=D_F(\omega) +C\delta(\omega-\Omega)+D\delta(\omega+\Omega).$$

Their coefficients encode the state and boundary condition. The differential operator fixes the response away from its characteristic frequencies, but it cannot determine the population already present on shell.

The equation $M(\omega^2-\Omega^2)D(\omega)=i$ fixes the off-shell part of the Green function but not the on-shell homogeneous terms. The pole prescription and the state determine which Green function is meant.

For a number eigenstate or a diagonal density matrix with

$$\langle a^\dagger a\rangle=n, \qquad \langle aa^\dagger\rangle=1+n,$$

one finds

$$\langle q(t)q(t')\rangle ={1\over2M\Omega}\big[(1+n)e^{-i\Omega(t-t')} +n e^{i\Omega(t-t')}\big] \qquad (t>t'),$$

and hence

$$D_n(t,t') ={1\over2M\Omega}\left[(1+n)e^{-i\Omega|t-t'|} +n e^{i\Omega|t-t'|}\right].$$

The ground-state result is recovered at $n=0$. A thermal state has

$$n={1\over e^{\beta\Omega}-1}.$$

Thus a Green function is not merely the inverse of a differential operator. It is the inverse plus a boundary condition plus a state.

Time-dependent backgrounds and the road to pair creation

The next page studies backgrounds that change in time. In oscillator language the mode equation becomes

$$\left({d^2\over dt^2}+\Omega_0^2+U(t)\right)f(t)=0,$$

with, for example,

$$U(t\to-\infty)=0, \qquad U(t\to+\infty)=\text{constant}.$$

A positive-frequency mode in the far past need not remain purely positive-frequency in the far future. This is the origin of Bogoliubov coefficients and particle creation:

$$f_{\rm in}(t)=\alpha f_{\rm out}(t)+\beta f_{\rm out}^*(t).$$

The in–out propagator uses boundary data at both ends: for fixed second time it is negative-frequency as the first time tends to the far past and positive-frequency as that time tends to the far future. Equivalently, its two homogeneous solutions are selected by the in- and out-vacuum boundary conditions. The in–in propagators are instead fixed by an initial state and track what that state evolves into. For stable scattering in empty space, both languages coexist peacefully. For particle production, vacuum decay, transport, and horizons, the distinction becomes the calculation.

Summary

The ordinary Feynman functional is an in–out object. It computes transition amplitudes and the in–out effective action. Its vacuum bubbles exponentiate into the vacuum persistence amplitude $S_{00}=e^{iW}$, and an imaginary part of $W$ signals loss of probability from the vacuum channel.

The Schwinger–Keldysh formalism is an in–in construction. It doubles fields, sources, vertices, and propagators because it evolves both the ket and the bra. For a normalized initial density matrix and unitary evolution, equal sources give $Z_{\rm SK}[J,J]=1$. In the Keldysh rotation, differentiating with respect to the auxiliary source $J_q$ inserts an observable, whereas varying the common source $J_c$ produces its physical response.

The four real-time propagators are Feynman, anti-Feynman, and two Wightman functions. In the $c/q$ basis they become the statistical function $F$, the retarded and advanced functions, and a vanishing $qq$ entry. Causal response appears when the two contour branches combine time-ordered terms into commutators.

A Green function is not just an inverse operator. It also contains a pole prescription and state data. That fact is harmless in equilibrium, but indispensable in nonequilibrium QFT.

Common pitfalls

Confusing Feynman propagation with causal response. The Feynman propagator is generally nonzero at spacelike separation. Causality is carried by commutators and retarded functions, not by the support of $D_{++}$ alone.

Calling an in–out matrix element an expectation value. A denominator-normalized Feynman expression still has a final-vacuum boundary condition. A detector reading at time $t$ is an in–in expectation value in the evolved initial state.

Dropping the minus-branch sign. The $-$ branch carries $-iS[\phi_-]$, so its vertices have conjugate signs. Without this sign, the two evolutions are not $U$ and $U^\dagger$, and vacuum-bubble cancellation fails.

Forgetting the initial density matrix. The contour defines an evolution problem only after $\rho_0$ is specified. Non-Gaussian initial correlations can add vertices at $t_0$ and cannot in general be reproduced by changing a free occupation number.

Using vacuum Wightman functions for a populated state. The displayed $2\pi\theta(\pm p^0)\delta(p^2-m^2)$ formulas are vacuum formulas. Thermal or occupied Gaussian states add occupation factors even though the free spectral support remains on shell.

Thinking the differential equation fixes the propagator. It fixes the off-shell inverse. The pole prescription and homogeneous on-shell terms encode boundary conditions and state data.

Exercises

Exercise 1: First-order causal response

Derive the first-order in–in response formula

$$\delta\langle B(t)\rangle =i\int_{-\infty}^{t}dt'\,\langle[H_I(t'),B_I(t)]\rangle_0.$$

Solution

Use

$$S(t,-\infty)=1-i\int_{-\infty}^t dt'\,H_I(t')+O(H_I^2),$$

and

$$S^\dagger(t,-\infty)=1+i\int_{-\infty}^t dt'\,H_I(t')+O(H_I^2).$$

Then

$$\langle B(t)\rangle =\langle0|S^\dagger(t,-\infty)B_I(t)S(t,-\infty)|0\rangle.$$

Keeping terms through first order gives

$$\langle B(t)\rangle =\langle B_I(t)\rangle_0 +i\int_{-\infty}^t dt'\,\langle H_I(t')B_I(t)\rangle_0 -i\int_{-\infty}^t dt'\,\langle B_I(t)H_I(t')\rangle_0.$$

Thus

$$\delta\langle B(t)\rangle =i\int_{-\infty}^{t}dt'\,\langle[H_I(t'),B_I(t)]\rangle_0.$$

The upper limit and the commutator are the two ingredients that make the expression retarded.

Exercise 2: Contour identities

Starting from

$$D_{++}=\theta(t)D_{-+}+\theta(-t)D_{+-}, \qquad D_{--}=\theta(-t)D_{-+}+\theta(t)D_{+-},$$

where $t=x^0-y^0$, prove

$$D_{++}+D_{--}=D_{+-}+D_{-+}$$

and

$$D_R=D_{++}-D_{+-}=D_{-+}-D_{--}.$$

Solution

Adding the two definitions gives

$$D_{++}+D_{--} =\big[\theta(t)+\theta(-t)\big]D_{-+} +\big[\theta(-t)+\theta(t)\big]D_{+-}.$$

Away from coincident times, $\theta(t)+\theta(-t)=1$, so

$$D_{++}+D_{--}=D_{-+}+D_{+-}.$$

For the retarded combination,

$$D_{++}-D_{+-} =\theta(t)D_{-+}+\theta(-t)D_{+-}-D_{+-} =\theta(t)(D_{-+}-D_{+-}).$$

This is

$$D_R(x,y)=\theta(x^0-y^0)\langle[\phi(x),\phi(y)]\rangle.$$

Similarly,

$$D_{-+}-D_{--} =D_{-+}-\theta(-t)D_{-+}-\theta(t)D_{+-} =\theta(t)(D_{-+}-D_{+-}),$$

so both expressions agree.

Exercise 3: Occupied oscillator propagator

For

$$q(t)={1\over\sqrt{2M\Omega}} (ae^{-i\Omega t}+a^\dagger e^{i\Omega t}),$$

assume a diagonal state with

$$\langle a^\dagger a\rangle=n, \qquad \langle aa^\dagger\rangle=1+n.$$

Compute $D_>(t,t')=\langle q(t)q(t')\rangle$, $D_<(t,t')=\langle q(t')q(t)\rangle$, and the time-ordered propagator $D_n(t,t')$.

Solution

Multiplying the two oscillator expansions gives

$$\langle q(t)q(t')\rangle ={1\over2M\Omega}\left[ \langle aa^\dagger\rangle e^{-i\Omega t}e^{i\Omega t'} +\langle a^\dagger a\rangle e^{i\Omega t}e^{-i\Omega t'} \right].$$

Therefore

$$D_>(t,t') ={1\over2M\Omega}\left[(1+n)e^{-i\Omega(t-t')} +n e^{i\Omega(t-t')}\right].$$

Interchanging $t$ and $t'$ gives

$$D_<(t,t') ={1\over2M\Omega}\left[(1+n)e^{i\Omega(t-t')} +n e^{-i\Omega(t-t')}\right].$$

The time-ordered function is

$$D_n(t,t')=\theta(t-t')D_>(t,t')+\theta(t'-t)D_<(t,t').$$

Combining the two cases yields

$$D_n(t,t') ={1\over2M\Omega}\left[(1+n)e^{-i\Omega|t-t'|} +n e^{i\Omega|t-t'|}\right].$$

For $n=0$, this reduces to the ground-state Feynman function.

Exercise 4: Closed-contour normalization

Show that the closed-time-path generating functional satisfies

$$Z_{\rm SK}[J,J]=1$$

for a normalized initial density matrix, while the in–out vacuum functional need not have unit modulus.

Solution

By definition,

$$Z_{\rm SK}[J_+,J_-] =\operatorname{Tr}\left[U_{J_+}\rho_0 U_{J_-}^\dagger\right].$$

Set $J_+=J_-=J$. Then

$$Z_{\rm SK}[J,J] =\operatorname{Tr}\left[U_J\rho_0 U_J^\dagger\right].$$

Using cyclicity of the trace,

$$\operatorname{Tr}(U_J\rho_0U_J^\dagger) =\operatorname{Tr}(U_J^\dagger U_J\rho_0).$$

Unitary time evolution gives $U_J^\dagger U_J=1$, hence

$$Z_{\rm SK}[J,J]=\operatorname{Tr}\rho_0=1.$$

The in–out functional is instead

$$Z_{\rm in\text{-}out}[J]=\langle0_{\rm out}|0_{\rm in}\rangle_J.$$

This is a transition amplitude between specified boundary states. Its modulus can be smaller than one when other final states are available.

Exercise 5: Microcausality and response support

Let

$$H_I(t)=\int d^{d-1}\mathbf x\,{\lambda\over4!}\phi^4(t,\mathbf x).$$

Use the first-order response formula to show that the correction to $\langle\phi^2(0)\rangle$ can be written as an integral over a commutator, and explain why only the past light cone of the origin contributes.

Solution

Set

$$B(0)=\phi^2(0).$$

The first-order in–in response is

$$\delta\langle\phi^2(0)\rangle =i\int_{-\infty}^{0}dt\,\langle[H_I(t),\phi^2(0)]\rangle_0.$$

Substituting the interaction Hamiltonian gives

$$\delta\langle\phi^2(0)\rangle =i{\lambda\over4!}\int_{-\infty}^{0}dt\int d^{d-1}\mathbf x\, \langle[\phi^4(t,\mathbf x),\phi^2(0)]\rangle_0.$$

Equivalently,

$$\delta\langle\phi^2(0)\rangle =i{\lambda\over4!}\int_{x^0<0}d^dx\, \langle[\phi^4(x),\phi^2(0)]\rangle_0.$$

For local relativistic fields, operators commute at spacelike separation. Therefore

$$[\phi^4(x),\phi^2(0)]=0 \qquad\text{when }x^2<0.$$

The time restriction $x^0<0$ already keeps only the past half-space, and microcausality further restricts the support to the past light cone.

Exercise 6: The Keldysh rotation

Use the definitions

$$\phi_c={\phi_++\phi_-\over2}, \qquad \phi_q=\phi_+-\phi_-,$$

and

$$J_c={J_++J_-\over2}, \qquad J_q=J_+-J_-,$$

to show that

$$J_+\phi_+-J_-\phi_-=J_c\phi_q+J_q\phi_c.$$

Explain why equal physical sources correspond to $J_q=0$. Then use the four contour propagators to show

$$\langle\phi_c\phi_c\rangle=F, \qquad \langle\phi_c\phi_q\rangle=D_R, \qquad \langle\phi_q\phi_c\rangle=D_A, \qquad \langle\phi_q\phi_q\rangle=0.$$

Solution

Invert the definitions:

$$\phi_+=\phi_c+{1\over2}\phi_q, \qquad \phi_-=\phi_c-{1\over2}\phi_q,$$

and

$$J_+=J_c+{1\over2}J_q, \qquad J_-=J_c-{1\over2}J_q.$$

Then

$$J_+\phi_+-J_-\phi_- =\left(J_c+{J_q\over2}\right)\left(\phi_c+{\phi_q\over2}\right) -\left(J_c-{J_q\over2}\right)\left(\phi_c-{\phi_q\over2}\right).$$

The $J_c\phi_c$ and $J_q\phi_q/4$ terms cancel, leaving

$$J_+\phi_+-J_-\phi_-=J_c\phi_q+J_q\phi_c.$$

Equal physical sources mean $J_+=J_-=J$, so

$$J_c=J, \qquad J_q=0.$$

The auxiliary source $J_q$ is nevertheless useful because differentiating with respect to it inserts the physical field $\phi_c$ before the physical limit is taken.

For the correlators,

$$\langle\phi_c\phi_c\rangle ={1\over4}(D_{++}+D_{+-}+D_{-+}+D_{--}) ={1\over2}(D_{+-}+D_{-+})=F,$$

where the contour identity was used in the second step. Similarly,

$$\langle\phi_c\phi_q\rangle ={1\over2}(D_{++}-D_{+-}+D_{-+}-D_{--}) =D_R,$$

and

$$\langle\phi_q\phi_c\rangle ={1\over2}(D_{++}+D_{+-}-D_{-+}-D_{--}) =D_A.$$

Finally,

$$\langle\phi_q\phi_q\rangle =D_{++}-D_{+-}-D_{-+}+D_{--}=0.$$

Thus the $c/q$ basis separates the statistical, retarded, and advanced functions and makes the unitarity zero explicit.

References

• Richard P. Feynman and Frank L. Vernon Jr., “The Theory of a General Quantum System Interacting with a Linear Dissipative System,” Annals of Physics 24 (1963), 118–173.
• Leonid V. Keldysh, “Diagram Technique for Nonequilibrium Processes,” Soviet Physics JETP 20 (1965), 1018–1026.
• Julian Schwinger, “Brownian Motion of a Quantum Oscillator,” Journal of Mathematical Physics 2 (1961), 407–432.

Further reading

• Esteban A. Calzetta and Bei-Lok B. Hu, Nonequilibrium Quantum Field Theory, Cambridge University Press, 2008.
• Alex Kamenev, Field Theory of Non-Equilibrium Systems, Cambridge University Press, 2011.
• Jørgen Rammer, Quantum Field Theory of Non-equilibrium States, Cambridge University Press, 2007.