nLab microcausal polynomial observable

Idea

The microcausal functionals on the space $C^\infty(X)$ of smooth functions on a globally hyperbolic spacetime $(X,e)$ are those which come from compactly supported distributions on some Cartesian product of copies of $X$ such that the wave front set of the distributions excludes those covectors to a point in $X^n$ all whose components are in the closed future cone or all whose components are in the closed past cone

These functionals underly the Wick algebra of free field theories. The condition on the wave front is such that the product of distributions with a Hadamard distribution is well defined, so that the coresponding Moyal star product is well defined, which gives the Wick algebra. At the same time the condition includes local observables and hence in particular the usual (adiabatically switched) point-interaction terms, such as of phi^4 theory.

types of observables in perturbative quantum field theory:

$$\array{ && \text{local} \\ && & \searrow \\ \text{field} &\longrightarrow& \text{linear} &\longrightarrow& \text{microcausal} &\longrightarrow& \text{polynomial} &\longrightarrow& \text{general} \\ && & \nearrow \\ && \text{regular} }$$

Definition

Definition

(polynomial observable)

Let $E \overset{fb}{\to}$ be field bundle which is a vector bundle. An off-shell polynomial observable is a smooth function

$$A \;\colon\; \Gamma_\Sigma(E) \longrightarrow \mathbb{C}$$

on the on-shell space of sections of the field bundle $E \overset{fb}{\to} \Sigma$ (space of field histories) which may be expressed as

$$A(\Phi) \;=\; \alpha^{(0)} + \int_\Sigma \alpha^{(1)}_a(x) \Phi^a(x) \, dvol_\Sigma(x) + \int_\Sigma \int_\Sigma \alpha^{(2)}_{a_1 a_2}(x_1, x_2) \Phi^{a_1}(x_1) \Phi^{a_2}(x_2) \,dvol_\Sigma(x_1) \, dvol_\Sigma(x_2) + \cdots \,,$$

where

$$\alpha^{(k)} \in \Gamma'_{\Sigma^k}\left((E^\ast)^{\boxtimes^k_{sym}} \right)$$

is a compactly supported distribution of k variables on the $k$-fold graded-symmetric external tensor product of vector bundles of the field bundle with itself.

Write

$$PolyObs(E) \hookrightarrow Obs(E)$$

for the subspace of off-shell polynomial observables onside all off-shell observables.

Let moreover $(E,\mathbf{L})$ be a free Lagrangian field theory whose equations of motion are Green hyperbolic differential equations. Then an on-shell polynomial observable is the restriction of an off-shell polynomial observable along the inclusion of the on-shell space of field histories $\Gamma_{\Sigma}(E)_{\delta_{EL}\mathbf{L} = 0} \hookrightarrow \Gamma_\Sigma(E)$. Write

$$PolyObs(E,\mathbf{L}) \hookrightarrow Obs(E,\mathbf{L})$$

for the subspace of all on-shell polynomial observables inside all on-shell observables.

By this prop. restriction yields an isomorphism between polynomial on-shell observables and polynomial off-shell observables modulo the image of the differential operator $P$:

$$PolyObs(E,\mathbf{L}) \underoverset{\simeq}{\text{restriction}}{\longleftarrow} PolyObs(E)/im(P) \,.$$

Definition

(microcausal observable)

For $\Sigma$ a spacetime, hence a Lorentzian manifold with time orientation, then a microcausal observable is a polynomial observable (def. ) such that each coefficient $\alpha^{(k)}$ has wave front set excluding those points where all $k$ wave vectors are in the closed future cone or all in the closed past cone.

Examples

Example

(non-singular observables are microcausal)

Let $(E,\mathbf{L})$ be a free Lagrangian field theory.

Then a regular observable, hence a polynomial observable (this def.) whose distributional coefficients $\alpha_{a_1 \cdots a_k}$ (?) are non-singular distributions is a microcausal observable (def. ).

This is simply because the wave front set of non-singular distributions is empty (by definition, via the Paley-Wiener-Schwartz theorem, this prop.).

Example

(compactly averaged point evaluations are microcausal)

Let $(E,\mathbf{L})$ be a free Lagrangian field theory. Assume the field bundle $E$ is a trivial vector bundle with linear fiber coordinates $(\phi^a)$.

Let $g \in C^\infty_c(X)$ be a bump function, then for $n \in \mathbb{N}$ the polynomial observables (this def.) of the form

$$\array{ \Gamma_\Sigma(E) &\overset{}{\longrightarrow}& \mathbb{C} \\ \Phi &\mapsto& \int_X g(x) \tilde \alpha_{a_1 \cdots a_k}(x) \Phi^{a_1}(x) \cdots \Phi^(a_k)\, dvol_\Sigma(x) }$$

are microcausal (def. ).

If here we think of $\phi(x)^n$ as a point-interaction term (as for instance in phi^4 theory) then $g$ is to be thought of as an “adiabatically switched” coupling constant. These are the relevant interaction terms to be quantized via causal perturbation theory.

Proof

For notational convenience, consider the case of the scalar field with $k = 2$; the general case is directly analogous. Then the local observable coming from $\phi^2$ (a phi^n interaction-term), has, regarded as a polynomial observable, the delta distribution $\delta(x_1-x_2)$ as coefficient in degree 2:

$$\begin{aligned} A(\Phi) & = \underset{\Sigma}{\int} g(x) (\Phi(x))^2 \,dvol_\Sigma(x) \\ & = \underset{\Sigma \times \Sigma}{\int} \underset{ = \alpha^{(2)}}{ \underbrace{ g(x_1) \delta(x_1 - x_2) }} \, \Phi(x_1) \Phi(x_2) \, dvol_\Sigma(x_1) \, dvol_\Sigma(x_2) \end{aligned} \,.$$

Now for $(x_1, x_2) \in \Sigma \times \Sigma$ and $\mathbb{R}^{2n} \simeq U \subset X \times X$ a chart around this point, the Fourier transform of distributions of $g \cdot \delta(-,-)$ restricted to this chart is proportional to the Fourier transform $\hat g$ of $g$ evaluated at the sum of the two covectors:

$$\begin{aligned} (k_1, k_2) & \mapsto \underset{\mathbb{R}^{2n}}{\int} g(x_1) \delta(x_1, x_2) e^{i (k_1 \cdot x_1 + k_2 \cdot x_2 )} \, dvol_\Sigma(x_1) \, dvol_\Sigma(x_2) \\ & \propto \hat g(k_1 + k_2) \end{aligned} \,.$$

Since $g$ is a plain bump function, its Fourier transform $\hat g$ is quickly decaying (according to this inequality) along $k_1 + k_2$ (this prop.), as long as $k_1 + k_2 \neq 0$. Only on the cone $k_1 + k_2 = 0$ the Fourier transform is constant, and hence in particular not decaying.

This means that the wave front set consists of the elements of the form $(x, (k, -k))$ with $k \neq 0$. Since $k$ and $-k$ are both in the closed future cone or both in the closed past cone precisely if $k = 0$, this situation is excluded in the wave front set and hence the distribution $g \cdot \delta(-,-)$ is microcausal.

(graphics grabbed from Khavkine-Moretti 14, p. 45)

This shows that microcausality in this case is related to conservation of momentum in the point interaction.

More generally:

Example

(polynomial local observables are microcausal)

Write

$$\Omega_{poly}^{h,v}(E)$$

for the space of differential forms on the jet bundle of the field bundle $E$ which locally are polynomials in the field variables.

$$\mathcal{F}_{loc} \; \subset \; C^\infty_c(\Sigma) \underset{\Omega_{poly}^{0,0}(E)}{\otimes} \Omega_{poly}^{d,0}(E)$$

for the subspace of horizontal differential forms of degree $d$ on the jet bundle (local Lagrangian densities) of those which are compactly supported with respect to $\Sigma$ (local observables) and polynomial with respect to the field variables.

Every $L \in \mathcal{F}_{loc}$ induces a functional

$$\Gamma_\Sigma(E) \longrightarrow \mathbb{R}$$

by integration of the pullback of $L$ along the jet prolongation of a given section:

$$\phi \mapsto \int_{\Sigma} j^\infty(\phi)^\ast L \,.$$

These functionals happen to be microcausal, so that there is an inclusion

$$\mathcal{F}_{loc} \hookrightarrow \mathcal{F}_{mc}$$

into the space of microcausal functionals (e.g. Fredenhagen-Rejzner 12, p. 21). In fact this is a dense subspace inclusion (e.g. Fredenhagen-Rejzner 12, p. 23)

Properties

Proposition

Write $\mathcal{F}_{reg} \subset \mathcal{F}_{mc}$ for that subalgebra of the algebra of microcausal functionals whose coefficients are non-singular distributions.

Let

$$\langle -\rangle \;\colon\; \mathcal{F}_{reg} \longrightarrow \mathbb{C}$$

be a state on regular observables which is quasi-free Hadamard. Then this uniquely extends to a state on microcausal functionsal

$$\array{ \mathcal{F}_{reg} &\overset{\langle -\rangle}{\longrightarrow}& \mathbb{C} \\ \downarrow & \nearrow_{\mathrlap{\exists ! \langle -\rangle}} \\ \mathcal{F}_{mc} }$$

(Hollands-Ruan 01, remark 1 on p. 12, implied by Brunetti-Fredenhagen 00, Hollands-Wald 01, a special case of Hollands-Ruan 01, theorem III.1 (ii))

Related concepts

quantum probability theory – observables and states

• states

  • classical state

  • quantum state

    • space of states (in geometric quantization)

    • state on a star-algebra, quasi-state

    • qbit, Bell state

      dimer, tensor network state

    • quantum state preparation

    • probability amplitude, quantum fluctuation

    • pure state

      • wave function

      • bra-ket

      • Bell state

    • quantum superposition, quantum interference

    • quantum entanglement

    • quantum measurement

      wave function collapse

      Born rule

      deferred measurement principle

      quantum reader monad

      measurement problem

    • superselection sector

    • mixed state, density matrix

      entanglement entropy

      holographic entanglement entropy

    • coherent quantum state

  • ground state, excited state

  • quasi-free state

  • Fock space, second quantization

  • vacuum, vacuum state

    • Hadamard state

    • vacuum diagram

    • vacuum expectation value, vacuum amplitude, vacuum fluctuation

    • vacuum energy

    • vacuum polarization

    • interacting vacuum

    • thermal vacuum, KMS state

    • vacuum stability

      • false vacuum, tachyon, Coleman-De Luccia instanton

    • theta vacuum

    • perturbative string theory vacuum

      • non-geometric string theory vacuum

      • landscape of string theory vacua

  • entangled state

  • tensor network state

    • matrix product state

    • tree tensor network state

• observables

  • quantum observable, beable

  • algebra of observables, star-algebra

  • Bohr topos

  • quantum operator (in geometric quantization)

  • quantum operation, quantum effect, effect algebra

  • in quantum field theory

    • local observable

    • polynomial observable

      • linear observable

        • field observable

    • regular observable

    • microcausal observable

    • normal-ordered product, time-ordered products, retarded product

    • Wick algebra

    • scattering amplitude

    • interacting field algebra of observables, Bogoliubov's formula

• GNS construction

• theorems

  • order-theoretic structure in quantum mechanics

    • Gleason's theorem

    • Alfsen-Shultz theorem

    • Harding-Döring-Hamhalter theorem

    • Kochen-Specker theorem

    • Nuiten's lemma

  • Wigner's theorem

  • no-cloning theorem

  • Bell's theorem

References

Original articles

• Romeo Brunetti, Klaus Fredenhagen, Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds, Commun. Math. Phys. 208 : 623-661, 2000 (math-ph/9903028)

• Michael Dütsch, Klaus Fredenhagen, Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion, Commun.Math.Phys. 219 (2001) 5-30 (arXiv:hep-th/0001129)

• Stefan Hollands, Robert Wald, Local Wick polynomials and time ordered products of quantum fields in curved spacetime, Commun. Math. Phys., Commun.Math.Phys.223:289-326,2001 (arXiv:gr-qc/0103074)

• Michael Dütsch, Klaus Fredenhagen, Perturbative algebraic field theory, and deformation quantization, in Roberto Longo (ed.), Mathematical Physics in Mathematics and Physics, Quantum and Operator Algebraic Aspects, volume 30 of Fields Institute Communications, pages 151–160. American Mathematical Society, 2001

  (arXiv:hep-th/0101079)

• Stefan Hollands, Weihua Ruan, The State Space of Perturbative Quantum Field Theory in Curved Spacetimes, Annales Henri Poincare 3 (2002) 635-657 (arXiv:gr-qc/0108032)

• Romeo Brunetti, Klaus Fredenhagen, Pedro Lauridsen Ribeiro, section 4.1 of Algebraic Structure of Classical Field Theory: Kinematics and Linearized Dynamics for Real Scalar Fields (arXiv:1209.2148)

Review

• Klaus Fredenhagen, Katarzyna Rejzner, Perturbative algebraic quantum field theory, In Mathematical Aspects of Quantum Field Theories, Springer 2016 (arXiv:1208.1428)

• Igor Khavkine, Valter Moretti, Algebraic QFT in Curved Spacetime and quasifree Hadamard states: an introduction, Chapter 5 in Romeo Brunetti et al. (eds.) Advances in Algebraic Quantum Field Theory, Springer, 2015 (arXiv:1412.5945)

• Katarzyna Rejzner, section 4.4.1 of Perturbative Algebraic Quantum Field Theory, Mathematical Physics Studies, Springer 2016 (web)

• Michael Dütsch, def. 1.2 and equation (2.47) in From classical field theory to perturbative quantum field theory, 2018