The Peierls bracket and the covariant phase space of an interacting field theory

Intuition Beginner

Imagine a field spread across spacetime — say the height of a vibrating membrane, but reaching everywhere and obeying a wave-like law. The actual motions it can make are the ones that solve its equation. There is a whole space of such allowed motions, one for each way of starting it off. We want a way to ask: if I nudge the field by paying attention to one quantity, how does a second quantity change in response? The answer is a kind of paired sensitivity between any two measurements, and it is the heart of how a classical field becomes a quantum one.

The old recipe for that paired sensitivity splits spacetime into space and time, freezes one instant, and reads off positions and velocities on that instant. It works, but it hides the even-handed treatment of space and time that the field's own law respects. Picking a special instant feels like cheating on a theory built to look the same to every observer.

Peierls found a way to skip the special instant entirely. Pick a quantity built from the field. Use it to give the field's law a small extra push. That push spreads outward in two ways: forward in time, like ripples after a stone is dropped, and backward in time, like ripples converging onto where a stone will land. The forward response and the backward response are each a sensible motion of the field; their difference is what matters.

The difference between forward-only and backward-only response of one quantity, measured against a second quantity, is the paired sensitivity we wanted. It never mentions a clock, a slice, or a split. It uses only the field's equation and the even-handed contrast between cause-going-forward and cause-going-backward.

Visual Beginner

Picture a single spacetime diagram. Time runs up the page, space runs across. In the middle sits a small smudge: the place where we add a tiny push to the field's law, a push shaped by some chosen quantity. From that smudge two cones open.

The upward cone is the forward response. It is how the field rearranges itself in the future because of the push — the retarded answer, the one a cause is allowed to have. The downward cone is the backward response, the advanced answer, where the field rearranges in the past as if anticipating the push.

Neither cone alone is the paired sensitivity. The construction takes the forward response and subtracts the backward response. What survives is supported across the whole causal region joined to the smudge, and it is automatically even-handed: swapping the two quantities flips its sign. That sign flip is the picture's main lesson, and it is why the construction can stand in for a bracket.

Worked example Beginner

Take the simplest field: a massless wave on flat spacetime in one space and one time direction, obeying the wave equation. Forget fields for a moment and think of a single push at the origin. The forward (retarded) response of a wave to a sharp push at the origin is a disturbance that lives on and inside the future light cone — it is nonzero only at later times, spreading out at the speed of light. The backward (advanced) response is the mirror image, living on and inside the past light cone.

Now put numbers on the supports. Suppose the push happens at time zero and position zero. The forward response is nonzero only for time greater than zero, within distance equal to the elapsed time. At time equal to three, it reaches out to position three on each side. The backward response is nonzero only for time less than zero, again within distance equal to the elapsed time; at time equal to minus three it reaches position three on each side.

Subtract them. The forward-minus-backward combination is nonzero in both cones and vanishes everywhere outside the light cone of the origin — at, say, time zero and position five, both responses are zero, so the difference is zero. This vanishing outside the light cone is the seed of the rule that measurements made far apart in space, at the same time, do not disturb each other. The same forward-minus-backward recipe, applied to two field quantities instead of one push, gives their paired sensitivity.

Check your understanding Beginner

Formal definition Intermediate+

Throughout, $(M,g)$ is a smooth four-dimensional time-oriented globally hyperbolic Lorentzian manifold in signature $(-,+,+,+)$, as in 13.09.02. A classical field theory is specified by a configuration space $\mathcal{C}=C^{\infty }(M)$ (a real scalar field, for definiteness; tensor and gauge fields are handled by carrying indices) and a local action functional $S:\mathcal{C}\to \mathbb{R}$ of the form $S[\varphi ]={\int }_M\mathcal{L}(\varphi ,\nabla \varphi )d{\mathrm{vol}}_g$, formal where the integral is divergent but with well-defined variational derivatives, as in 09.02.02. The field equation is the Euler-Lagrange equation $S^{\prime }[\varphi ]=0$, where $S^{\prime }[\varphi ]\in {\mathcal{C}}^{\prime }$ is the first variation. The solution manifold (the on-shell configuration space) is

$${\mathcal{E}}_S:=\{\varphi \in \mathcal{C}:S^{\prime }[\varphi ]=0\}.$$

An observable is a (suitably regular) functional $A:\mathcal{C}\to \mathbb{R}$; what matters is its restriction to ${\mathcal{E}}_S$. The model case is a regular local functional $A[\varphi ]={\int }_{M}a(x)F(\varphi (x),\nabla \varphi (x))d{\mathrm{vol}}_g$ with $a\in C_c^{\infty }(M)$ a smooth compactly supported weight.

Fix a background solution $\varphi \in {\mathcal{E}}_S$. The second variation of the action at $\varphi$ is the linear differential operator

$$S^{\prime \prime }[\varphi ]:C^{\infty }(M)\to C_c^{\infty }(M{)}^{\prime },(S^{\prime \prime }[\varphi ]\psi )(x)={\int }_M\frac{{\delta }^{2}S}{\delta \varphi (x)\delta \varphi (y)}{|}_{\varphi }\psi (y)d{\mathrm{vol}}_g(y),$$

the Jacobi operator (or linearised field operator) of the theory at $\varphi$. Its kernel is the space of on-shell linearised solutions — the tangent space $T_{\varphi }{\mathcal{E}}_S$. We assume $S^{\prime \prime }[\varphi ]$ is normally hyperbolic for every $\varphi \in {\mathcal{E}}_S$ of interest: its principal symbol is $g^{\mu \nu }{\xi }_{\mu }{\xi }_{\nu }$ times the identity. For the free Klein-Gordon action $S[\varphi ]=-\frac{1}{2}{\int }_M({\nabla }^{\mu }\varphi {\nabla }_{\mu }\varphi +m^2{\varphi }^2)d{\mathrm{vol}}_g$ one has $S^{\prime \prime }[\varphi ]=-({\square }_g+m^2)=-P$ independent of $\varphi$; for an interacting action $S[\varphi ]=S_{\mathrm{free}}[\varphi ]-\frac{\lambda }{4!}\int {\varphi }^4$ one finds $S^{\prime \prime }[\varphi ]=-({\square }_g+m^2+\frac{\lambda }{2}{\varphi }^2)$, an operator with a $\varphi$-dependent potential.

By the well-posedness theory of 13.09.02 applied to the normally hyperbolic operator $S^{\prime \prime }[\varphi ]$, there exist unique retarded and advanced Green's operators ${\tilde{G}}_{\varphi }^{\pm }:C_c^{\infty }(M)\to C^{\infty }(M)$ with $S^{\prime \prime }[\varphi ]{\tilde{G}}_{\varphi }^{\pm }=\mathrm{id}$, ${\tilde{G}}_{\varphi }^{\pm }{S}^{\prime \prime }[\varphi ]=\mathrm{id}$ on $C_c^{\infty }$, and $\mathrm{supp}({\tilde{G}}_{\varphi }^{\pm }f)\subset J^{\pm }(\mathrm{supp}f)$. Their difference is the causal Green's operator (the Pauli-Jordan operator of the linearised theory at $\varphi$):

$${\tilde{G}}_{\varphi }:={\tilde{G}}_{\varphi }^{-}-{\tilde{G}}_{\varphi }^{+}:C_c^{\infty }(M)\to C^{\infty }(M),$$

with antisymmetric distributional kernel and image inside $\ker S^{\prime \prime }[\varphi ]=T_{\varphi }{\mathcal{E}}_S$. At the free Klein-Gordon point this is exactly $E=E^{-}-E^{+}$ of 13.09.02, up to the sign of $S^{\prime \prime }=-P$.

For an observable $A$ regard $A^{\prime }[\varphi ]\in C_c^{\infty }(M)$ as a source (the integral kernel of the first variation; compactly supported when $A$ is a regular local functional). The retarded and advanced effects of $A$ on a second observable $B$ at the solution $\varphi$ are the directional derivatives of $B$ along the on-shell linearised response of the field to perturbing the action $S\mapsto S+ϵA$:

$$D_A^{\pm }B[\varphi ]:={\int }_M{B}^{\prime }[\varphi ](x)({\tilde{G}}_{\varphi }^{\pm }{A}^{\prime }[\varphi ])(x)d{\mathrm{vol}}_g(x)=⟨B^{\prime }[\varphi ],{\tilde{G}}_{\varphi }^{\pm }{A}^{\prime }[\varphi ]⟩.$$

The Peierls bracket of $A$ and $B$ at $\varphi$ is

$$\boxed{\{A,B\}[\varphi ]:=D_A^{-}B[\varphi ]-D_A^{+}B[\varphi ]=⟨B^{\prime }[\varphi ],{\tilde{G}}_{\varphi }{A}^{\prime }[\varphi ]⟩.}$$

This is DeWitt's manifestly covariant Poisson bracket on the solution manifold ${\mathcal{E}}_S$ ^{[DeWitt 2003]}. It depends on a choice of spacetime orientation (which Green's operator is "retarded") but on no spatial slice. For the free field with $A=\hat{\varphi }(f)$, $B=\hat{\varphi }(h)$ the smeared field functionals $\hat{\varphi }(f)[\varphi ]=\int f\varphi d{\mathrm{vol}}_g$, the bracket evaluates to the slice-independent number $\{\hat{\varphi }(f),\hat{\varphi }(h)\}=⟨f,Eh⟩=E(f,h)$, the Pauli-Jordan kernel of 13.09.02.

Key theorem with proof Intermediate+

Theorem (Antisymmetry and the canonical-bracket equivalence). Let $S^{\prime \prime }[\varphi ]$ be a formally self-adjoint normally hyperbolic Jacobi operator on the globally hyperbolic spacetime $(M,g)$, with causal Green's operator ${\tilde{G}}_{\varphi }={\tilde{G}}_{\varphi }^{-}-{\tilde{G}}_{\varphi }^{+}$. Then for regular local functionals $A,B$:

(i) The Peierls bracket is antisymmetric: $\{A,B\}[\varphi ]=-\{B,A\}[\varphi ]$.

(ii) On the free Klein-Gordon theory, with $A,B$ taken to be the field and its conjugate momentum smeared against test functions on a Cauchy slice $\Sigma$, the Peierls bracket equals the canonical equal-time Poisson bracket of 05.02.02 and 12.12.01: $\{\varphi (x),\pi (y){\}}_{\Sigma }={\delta }_{\Sigma }(x,y)$, with $\{\varphi ,\varphi {\}}_{\Sigma }=\{\pi ,\pi {\}}_{\Sigma }=0$.

Proof. (i) Formal self-adjointness of $S^{\prime \prime }[\varphi ]$ with respect to the $L^2(M,d{\mathrm{vol}}_g)$ pairing gives, exactly as in the mutual-adjointness proposition of 13.09.02, the relation $({\tilde{G}}_{\varphi }^{+}{)}^{\ast }={\tilde{G}}_{\varphi }^{-}$: for $f,h\in C_c^{\infty }(M)$,

$$⟨h,{\tilde{G}}_{\varphi }^{+}f⟩=⟨S^{\prime \prime }{\tilde{G}}_{\varphi }^{-}h,{\tilde{G}}_{\varphi }^{+}f⟩=⟨{\tilde{G}}_{\varphi }^{-}h,S^{\prime \prime }{\tilde{G}}_{\varphi }^{+}f⟩=⟨{\tilde{G}}_{\varphi }^{-}h,f⟩=⟨f,{\tilde{G}}_{\varphi }^{-}h⟩,$$

where the integration by parts has no boundary term because $\mathrm{supp}({\tilde{G}}_{\varphi }^{-}h)\cap \mathrm{supp}({\tilde{G}}_{\varphi }^{+}f)\subset J^{-}(\mathrm{supp}h)\cap J^{+}(\mathrm{supp}f)$ is compact by global hyperbolicity. Therefore ${\tilde{G}}_{\varphi }^{\ast }=({\tilde{G}}_{\varphi }^{-}-{\tilde{G}}_{\varphi }^{+}{)}^{\ast }={\tilde{G}}_{\varphi }^{+}-{\tilde{G}}_{\varphi }^{-}=-{\tilde{G}}_{\varphi }$, i.e. ${\tilde{G}}_{\varphi }$ has antisymmetric kernel. Writing $f=A^{\prime }[\varphi ]$, $h=B^{\prime }[\varphi ]$,

$$\{A,B\}[\varphi ]=⟨h,{\tilde{G}}_{\varphi }f⟩=⟨{\tilde{G}}_{\varphi }^{\ast }h,f⟩=-⟨{\tilde{G}}_{\varphi }h,f⟩=-⟨f,{\tilde{G}}_{\varphi }h⟩=-\{B,A\}[\varphi ].$$

(ii) Take the free action $S=-\frac{1}{2}\int (\nabla \varphi {)}^2+m^2{\varphi }^2$, so $S^{\prime \prime }=-({\square }_g+m^2)$ and ${\tilde{G}}_{\varphi }=-E$ with $E=E^{-}-E^{+}$ the Pauli-Jordan operator. For test functions $f,h\in C_c^{\infty }(M)$ the field functionals satisfy $\{\hat{\varphi }(f),\hat{\varphi }(h)\}=⟨h,{\tilde{G}}_{\varphi }f⟩=-⟨h,Ef⟩=-E(h,f)=E(f,h)$, using antisymmetry of $E$. Now choose a Cauchy slice $\Sigma$ and write the smeared Cauchy data: for the field $\varphi (\alpha ):={\int }_{\Sigma }\alpha \varphi d{\mathrm{vol}}_{\Sigma }$ and the momentum $\pi (\beta ):={\int }_{\Sigma }\beta n^{\mu }{\nabla }_{\mu }\varphi d{\mathrm{vol}}_{\Sigma }$ with $\alpha ,\beta \in C_c^{\infty }(\Sigma )$. The Cauchy-data form of the Pauli-Jordan kernel established in 13.09.02, ${\sigma }_{\Sigma }((u_0,u_1),(v_0,v_1))={\int }_{\Sigma }(u_1{v}_0-u_0{v}_1)d{\mathrm{vol}}_{\Sigma }$, gives, evaluated on solutions $u=Ef$, $v=Eh$ with the indicated Cauchy data,

$$\{\varphi (\alpha ),\pi (\beta )\}={\int }_{\Sigma }\alpha \beta d{\mathrm{vol}}_{\Sigma },\{\varphi (\alpha ),\varphi ({\alpha }^{\prime })\}=0,\{\pi (\beta ),\pi ({\beta }^{\prime })\}=0,$$

which is precisely the canonical equal-time Poisson bracket $\{\varphi (x),\pi (y)\}={\delta }_{\Sigma }(x,y)$ in smeared form. The slice $\Sigma$ was arbitrary, and the left-hand sides are manifestly slice-independent, so the canonical bracket — apparently slice-dependent — is in fact the restriction of the slice-free Peierls bracket. $\square$

Bridge. This equivalence is exactly the foundational reason the Peierls bracket deserves to be called a Poisson bracket at all: it builds toward the full covariant phase space because it shows the slice-free construction reproduces, with no choice of $\Sigma$, the canonical structure that the $3+1$ split of 12.12.01 obtains only after breaking covariance. The central insight is that the causal Green's operator ${\tilde{G}}_{\varphi }$ — retarded minus advanced — is the manifestly covariant integral kernel of the symplectic form, so the canonical bracket and the covariant bracket are dual descriptions of one geometric object. This is exactly the statement that the Pauli-Jordan kernel $E$ of 13.09.02, read on a slice, becomes the equal-time commutator; putting these together, the same identity appears again in the quantisation rule $[\hat{A},\hat{B}]=i\hslash \widehat{\{A,B\}}$ that turns the classical Peierls bracket into the field commutator, and it generalises without change of form to the interacting theory through the $\varphi$-dependent Jacobi operator $S^{\prime \prime }[\varphi ]$.

Exercises Intermediate+

Advanced results Master

The covariant phase space and its symplectic form. The solution manifold ${\mathcal{E}}_S$, equipped with the Peierls bracket, is the covariant phase space of the theory. Its (pre)symplectic form arises directly from the boundary term of the action variation ^{[Crnkovic-Witten 1987]}: writing $\delta S[\varphi ]={\int }_M{S}^{\prime }[\varphi ]\delta \varphi +{\int }_{\partial }\theta [\varphi ,\delta \varphi ]$, the boundary one-form $\theta$ defines the presymplectic potential, and $\omega =\delta \theta$ integrated over a Cauchy slice $\Sigma$ gives the symplectic form ${\Omega }_{\Sigma }$ on ${\mathcal{E}}_S$. Conservation of the symplectic current makes ${\Omega }_{\Sigma }$ Cauchy-independent, and the Poisson bracket associated to $\Omega$ is exactly the Peierls bracket: for the free field ${\Omega }^{-1}$ has integral kernel ${\tilde{G}}_{\varphi }=-E$, so $\{A,B\}={\Omega }^{-1}(\mathrm{d}A,\mathrm{d}B)$ reproduces $⟨B^{\prime },{\tilde{G}}_{\varphi }{A}^{\prime }⟩$. The covariant phase space is symplectomorphic to the canonical phase space $(T^{\ast }\mathcal{Q},{\Omega }_{\mathrm{can}})$ of 12.12.01 via the Cauchy-data map $\varphi \mapsto (\varphi {|}_{\Sigma },n^{\mu }{\nabla }_{\mu }\varphi {|}_{\Sigma })$, but unlike the canonical description it carries no choice of $\Sigma$ in its definition.

The interacting / nonlinear lift. For a nonlinear action the construction proceeds solution-by-solution. At each $\varphi \in {\mathcal{E}}_S$ the Jacobi operator $S^{\prime \prime }[\varphi ]$ is a $\varphi$-dependent normally hyperbolic operator, with its own retarded/advanced Green's operators ${\tilde{G}}_{\varphi }^{\pm }$ and causal operator ${\tilde{G}}_{\varphi }$. The Peierls bracket $\{A,B\}[\varphi ]=⟨B^{\prime }[\varphi ],{\tilde{G}}_{\varphi }{A}^{\prime }[\varphi ]⟩$ is then a function on ${\mathcal{E}}_S$, not a single number; the bracket of two observables is itself an observable, and antisymmetry, the Leibniz rule, and the Jacobi identity hold pointwise on ${\mathcal{E}}_S$. In the algebraic-QFT formulation ^{[Brunetti-Fredenhagen-Ribeiro 2019]} the regular functionals form a Poisson $\ast$-algebra under this bracket, and perturbative interacting QFT is built by deforming it: the time-ordered products and the Bogoliubov $S$-matrix are constructed from the free Peierls bracket, with the interaction entering through the formal series in $\lambda$ around a free background.

Microlocal regularity. The bracket $\{A,B\}=⟨B^{\prime },\tilde{G}{A}^{\prime }⟩$ is a pairing of distributions and need not be defined for arbitrary functionals: ${\tilde{G}}_{\varphi }$ is a distribution on $M\times M$ with wave-front set on the bicharacteristic relation (as in 13.09.02/13.09.03), so the pairing converges when the wave-front sets of $A^{\prime }[\varphi ]$ and $B^{\prime }[\varphi ]$ are suitably disjoint from it ^{[Khavkine 2014]}. Regular local functionals (smooth compactly supported sources) always pair, which is why they are the natural domain; the extension to broader classes requires the Hörmander wave-front-set criterion for multiplying distributions, the same microlocal control that governs Hadamard states in 13.09.03.

Synthesis. The Peierls bracket is the foundational reason the covariant and canonical formulations of a field theory are dual descriptions of one structure: the retarded-minus-advanced causal Green's operator ${\tilde{G}}_{\varphi }$ is the integral kernel of the symplectic form, and this is exactly the kernel $E=E^{-}-E^{+}$ that 13.09.02 builds for the free field and 13.09.03 reads as the symplectic form on Cauchy data. Putting these together, the central insight is that quantisation $\{A,B\}\mapsto \frac{1}{i\hslash }[\hat{A},\hat{B}]$ sends the Peierls bracket to the field commutator with no choice of slice, so the covariant phase space generalises the canonical phase space to the manifestly relativistic setting and is dual to it under the Cauchy-data symplectomorphism. The construction builds toward perturbative interacting QFT, where the free Peierls bracket organises the deformation into time-ordered products, and it appears again in the algebraic approach where the on-shell functionals form a Poisson algebra whose deformation quantisation is the interacting field algebra. The bridge from a Lagrangian to a quantum theory passes through this bracket: without it, the relativistic Poisson structure that turns into the commutator would be visible only after the covariance-breaking $3+1$ split, and the gauge-invariant content of 12.12.01 would have no slice-free home.

Full proof set Master

Proposition 1 (Image of ${\tilde{G}}_{\varphi }$ lies in the linearised solution space). For every $f\in C_c^{\infty }(M)$, $S^{\prime \prime }[\varphi ]{\tilde{G}}_{\varphi }f=0$; that is, $\mathrm{im}{\tilde{G}}_{\varphi }\subseteq \ker S^{\prime \prime }[\varphi ]=T_{\varphi }{\mathcal{E}}_S$.

Proof. By definition $S^{\prime \prime }[\varphi ]{\tilde{G}}_{\varphi }^{\pm }=\mathrm{id}$ on $C_c^{\infty }(M)$, so $S^{\prime \prime }[\varphi ]{\tilde{G}}_{\varphi }f=S^{\prime \prime }[\varphi ]({\tilde{G}}_{\varphi }^{-}-{\tilde{G}}_{\varphi }^{+})f=f-f=0$. The kernel of the linearised field operator is, by definition of the second variation, the tangent space to the solution manifold at $\varphi$, the space of on-shell linearised perturbations. Hence ${\tilde{G}}_{\varphi }f$ is an on-shell linearised solution for every source $f$. $\square$

Proposition 2 (Antisymmetry of the causal Green's operator). If $S^{\prime \prime }[\varphi ]$ is formally self-adjoint with respect to $d{\mathrm{vol}}_g$, then ${\tilde{G}}_{\varphi }$ has antisymmetric distributional kernel: ${\tilde{G}}_{\varphi }(x,y)=-{\tilde{G}}_{\varphi }(y,x)$, equivalently $\tilde G_\phi^ = -\tilde G_\phi$.*

Proof. As in Exercise 4, formal self-adjointness gives $⟨h,{\tilde{G}}_{\varphi }^{+}f⟩=⟨{\tilde{G}}_{\varphi }^{-}h,f⟩$ for $f,h\in C_c^{\infty }(M)$, the boundary term vanishing because $J^{-}(\mathrm{supp}h)\cap J^{+}(\mathrm{supp}f)$ is compact by global hyperbolicity. Thus $({\tilde{G}}_{\varphi }^{+}{)}^{\ast }={\tilde{G}}_{\varphi }^{-}$ and symmetrically $({\tilde{G}}_{\varphi }^{-}{)}^{\ast }={\tilde{G}}_{\varphi }^{+}$. Therefore ${\tilde{G}}_{\varphi }^{\ast }=({\tilde{G}}_{\varphi }^{-}{)}^{\ast }-({\tilde{G}}_{\varphi }^{+}{)}^{\ast }={\tilde{G}}_{\varphi }^{+}-{\tilde{G}}_{\varphi }^{-}=-{\tilde{G}}_{\varphi }$. In kernel terms this is ${\tilde{G}}_{\varphi }(x,y)=-{\tilde{G}}_{\varphi }(y,x)$. $\square$

Proposition 3 (The Peierls bracket is a Poisson bracket). On the algebra of regular local functionals on ${\mathcal{E}}_S$, the Peierls bracket $\{A,B\}=⟨B^{\prime },{\tilde{G}}_{\varphi }{A}^{\prime }⟩$ is bilinear, antisymmetric, satisfies the Leibniz rule in each slot, and satisfies the Jacobi identity. It therefore makes the regular on-shell functionals a Poisson algebra.

Proof. Bilinearity is immediate from linearity of $A\mapsto A^{\prime }[\varphi ]$ and of ${\tilde{G}}_{\varphi }$. Antisymmetry is Proposition 2 transcribed to functionals: $\{A,B\}=⟨B^{\prime },{\tilde{G}}_{\varphi }{A}^{\prime }⟩=⟨{\tilde{G}}_{\varphi }^{\ast }{B}^{\prime },A^{\prime }⟩=-⟨{\tilde{G}}_{\varphi }{B}^{\prime },A^{\prime }⟩=-⟨A^{\prime },{\tilde{G}}_{\varphi }{B}^{\prime }⟩=-\{B,A\}$. The Leibniz rule $\{A,BC\}=\{A,B\}C+B\{A,C\}$ follows because the first variation obeys the product rule, $(BC{)}^{\prime }[\varphi ]=B^{\prime }[\varphi ]C[\varphi ]+B[\varphi ]C^{\prime }[\varphi ]$, and the bracket is linear in the source $(BC{)}^{\prime }$. The Jacobi identity is the content of Exercise 7: writing each nested bracket out and using $\delta {\tilde{G}}_{\varphi }=-{\tilde{G}}_{\varphi }(S^{\prime \prime \prime }[\varphi ]){\tilde{G}}_{\varphi }$ for the variation of the Green's operator, the cyclic sum collects the third-variation $S^{\prime \prime \prime }[\varphi ]$ contracted against three causal operators into a totally antisymmetric expression that vanishes; the precise distributional argument is Marolf's ^{[Marolf 1994]}, and the original derivation is DeWitt's ^{[DeWitt 2003]}. Hence all four axioms of a Poisson algebra hold. $\square$

Proposition 4 (Equivalence with the canonical bracket, free field). For the free Klein-Gordon theory, the Peierls bracket of two field functionals equals the Pauli-Jordan pairing, $\{\hat{\varphi }(f),\hat{\varphi }(h)\}=E(f,h)$, and restricting to Cauchy-slice-localised functionals reproduces the canonical equal-time Poisson bracket $\{\varphi (x),\pi (y){\}}_{\Sigma }={\delta }_{\Sigma }(x,y)$ for every Cauchy slice $\Sigma$.

Proof. This is part (ii) of the Key theorem. With $S^{\prime \prime }=-({\square }_g+m^2)$ one has ${\tilde{G}}_{\varphi }=-E$, so $\{\hat{\varphi }(f),\hat{\varphi }(h)\}=⟨h,{\tilde{G}}_{\varphi }f⟩=-⟨h,Ef⟩=E(f,h)$ by antisymmetry of $E$. The Cauchy-data form ${\sigma }_{\Sigma }((u_0,u_1),(v_0,v_1))={\int }_{\Sigma }(u_1{v}_0-u_0{v}_1)d{\mathrm{vol}}_{\Sigma }$ of 13.09.02, evaluated on the solutions generated by $f$ and $h$, yields the canonical relations $\{\varphi (\alpha ),\pi (\beta )\}={\int }_{\Sigma }\alpha \beta$, $\{\varphi ,\varphi \}=\{\pi ,\pi \}=0$. Slice-independence of the left side, established by conservation of the symplectic current (Exercise 6), shows the apparently $\Sigma$-dependent canonical bracket is the restriction of the slice-free Peierls bracket. $\square$

Connections Master

• Klein-Gordon on a globally hyperbolic spacetime 13.09.02 supplies the free-field special case: its retarded and advanced Green's operators $E^{\pm }$, the causal propagator $E=E^{-}-E^{+}$, and the Cauchy-data symplectic form are exactly what the Peierls construction produces when the Jacobi operator $S^{\prime \prime }[\varphi ]=-({\square }_g+m^2)$ is the free Klein-Gordon operator. The present unit names that structure the Peierls bracket and lifts it from the linear field to a nonlinear action by replacing $-({\square }_g+m^2)$ with the background-dependent $S^{\prime \prime }[\varphi ]$.

• Hadamard states via the wave-front-set criterion 13.09.03 reads the causal propagator $E$ as the symplectic form on Cauchy-data phase space and controls its wave-front set on the bicharacteristic relation. That microlocal control is what makes the Peierls bracket $⟨B^{\prime },{\tilde{G}}_{\varphi }{A}^{\prime }⟩$ a convergent pairing of distributions for regular functionals, and it is the same control that governs the multiplication of distributions in the deformation to the interacting quantum algebra.

• Poisson bracket and Poisson manifold 05.02.02 is the abstract structure the Peierls bracket reproduces covariantly. Proposition 3 verifies the Poisson-algebra axioms — bilinearity, antisymmetry, Leibniz, Jacobi — on the on-shell functionals, exhibiting ${\mathcal{E}}_S$ with the Peierls bracket as a (typically infinite-dimensional) Poisson manifold whose symplectic leaves are the connected components of the solution space.

• Euler-Lagrange equations / classical field action 09.02.02 provides the action $S[\varphi ]$ that is being perturbed and whose stationary points form the solution manifold ${\mathcal{E}}_S$. The first variation $S^{\prime }[\varphi ]=0$ defines the field equation; the second variation $S^{\prime \prime }[\varphi ]$ is the Jacobi operator whose causal Green's operator is the kernel of the bracket; the boundary term of $\delta S$ gives the presymplectic potential of the covariant phase space.

• Canonical quantum field theory 12.12.01 is the slice-dependent formulation the Peierls bracket reproduces without a $3+1$ split. The Cauchy-data map is a symplectomorphism from the covariant phase space to the canonical phase space, and quantisation $\{A,B\}\mapsto \frac{1}{i\hslash }[\hat{A},\hat{B}]$ sends the Peierls bracket to the equal-time field commutator, recovering the canonical commutation relations in a manifestly covariant way.

• Quantum energy inequalities 13.09.11 and the renormalisation programme of the chapter rest on the same free-field symplectic data; the Peierls bracket is the classical Poisson structure whose deformation quantisation produces the field algebra on which those bounds and the Wick-polynomial constructions are formulated.

Historical & philosophical context Master

The construction originates with Rudolf Peierls in Proc. R. Soc. Lond. A 214 (1952) 143 ^{[Peierls 1952]}, who sought a definition of the field commutator that did not presuppose a canonical $3+1$ decomposition and that would apply uniformly to fields of any spin and to gauge theories. Peierls defined the commutator of two quantities through the retarded-minus-advanced effect of perturbing the action by one of them, and proved its equivalence with the canonical commutator in the cases then known. The bracket lay relatively dormant until Bryce DeWitt made it the cornerstone of his manifestly covariant formulation of quantum field theory, developed across decades of lectures and consolidated in The Global Approach to Quantum Field Theory, Vols. 1-2 (Oxford, 2003) ^{[DeWitt 2003]}, where Chs. 3-4 build the bracket from the Green's functions of the Jacobi operator $S^{\prime \prime }[\varphi ]$, prove antisymmetry and the Jacobi identity, and extend the construction to gauge theories via the gauge-fixed Jacobi operator and the associated ghost structure.

The mathematical consolidation came later. Donald Marolf in Ann. Phys. (NY) 236 (1994) 392 and 410 ^{[Marolf 1994]} gave careful proofs that the Peierls bracket is a Poisson bracket on the space of solutions, including the constrained and gauge cases, and clarified its relation to the bracket on the space of histories. The geometric covariant-phase-space picture, in which the symplectic form arises from the boundary term of the action variation, was developed by Čedomir Crnković and Edward Witten in Three Hundred Years of Gravitation (Cambridge, 1987) ^{[Crnkovic-Witten 1987]}. The modern microlocal and algebraic-QFT treatment, identifying the Peierls bracket as the Poisson structure of a $\ast$-algebra of regular functionals whose deformation quantisation yields perturbative interacting QFT, is due to Romeo Brunetti, Klaus Fredenhagen and Pedro Lauridsen Ribeiro ^{[Brunetti-Fredenhagen-Ribeiro 2019]} and reviewed comprehensively by Igor Khavkine in Int. J. Mod. Phys. A 29 (2014) 1430009 ^{[Khavkine 2014]}, who connects the Peierls formula to the covariant phase space, the constraint analysis, and the equivalence with the canonical bracket.

Bibliography Master

Originating papers:

• Peierls, R. E., "The commutation laws of relativistic field theory", Proc. R. Soc. Lond. A 214 (1952), 143-157. [Originating definition of the covariant commutator from the retarded-minus-advanced response of the action.]
• DeWitt, B. S., The Global Approach to Quantum Field Theory, Vols. 1-2 (Oxford University Press, 2003). [The definitive development of the Peierls bracket as the foundation of manifestly covariant QFT; Chs. 3-4.]

Mathematical structure of the bracket:

• Marolf, D., "Poisson brackets on the space of histories", Ann. Phys. (NY) 236 (1994), 392-412. [Proof that the Peierls bracket is a Poisson bracket; the space-of-histories viewpoint.]
• Marolf, D., "The generalized Peierls bracket", Ann. Phys. (NY) 236 (1994), 413-447. [The constrained / gauge case of the Peierls bracket.]
• Khavkine, I., "Covariant phase space, constraints, gauge and the Peierls formula", Int. J. Mod. Phys. A 29 (2014), 1430009. [Modern review tying the Peierls formula to the covariant phase space and the canonical bracket.]
• Forger, M. & Romero, S. V., "Covariant Poisson brackets in geometric field theory", Comm. Math. Phys. 256 (2005), 375-410. [Multisymplectic geometric formulation of the covariant Poisson bracket.]

Covariant phase space and algebraic QFT:

• Crnković, Č. & Witten, E., "Covariant description of canonical formalism in geometrical theories", in Three Hundred Years of Gravitation, eds. S. W. Hawking & W. Israel (Cambridge University Press, 1987), 676-684. [The covariant phase space and its presymplectic form from the action boundary term.]
• Brunetti, R., Fredenhagen, K. & Ribeiro, P. L., "Algebraic structure of classical field theory: kinematics and linearized dynamics for real scalar fields", Comm. Math. Phys. 368 (2019), 519-584 (arXiv:1306.1058). [The Peierls bracket in perturbative algebraic QFT; the on-shell functional algebra.]
• Wald, R. M., Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (University of Chicago Press, 1994). [§3.3: the symplectic structure on the free-field solution space the Peierls bracket reproduces in the linear case.]