algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
field theory: classical, pre-quantum, quantum, perturbative quantum
A Wick algebra is an algebra of quantum observables of free quantum fields.
In the quantum mechanics of harmonic oscillators or in quantum field theory of free fields on Minkowski spacetime one encounters linear operators $\{a_k, a^\ast_k\}_{k \in K}$ that satisfy the canonical commutation relation $[a_i, a^\ast_j] = diag((c_k))_{i j}$. Then by a normal ordered polynomial or Wick polynomial (Wick 50) one means a polynomial, denoted $:P:$, which is obtained from a polynomial $P((a_k, a^\ast_k))$ in these operators by ordering all “creation operators” $a_k^\ast$ to the left of all “annihiliation operators”. For example focusing on a single mode $k$ we have:
The intuitive idea is that these operators span a Hilbert space $\mathcal{H}$ of quantum states from a vacuum state $\vert vac \rangle \in \mathcal{H}$ characterized by the condition
hence (if we think of $a_k$ as acting by “removing a quantum in mode $k$”) by the condition that it contains no quanta. So the normal ordered Wick polynomials represent the quantum observables with vanishing vacuum expectation value. In quantum field theory they model scattering processes where quanta enter a reaction process (the modes corresponding to the “annihilation” operators $a_k$) and other particles come out of the reaction (the modes corresppnding to the “creation” operators $a^\ast_k$).
The product of two Wick polynomials, computed in the ambient operator algebra and then re-expressed as a Wick polynomial, is given by computing the relevant sequence of commutators by Wick's lemma, for example
where $c \coloneqq [a, a^\ast]$ is the value of the canonical commutator.
The associative algebra thus obtained is hence called the algebra of normal ordered operators or Wick polynomial algebra or just Wick algebra.
This plays a central role in perturbative quantum field theory, where the quantization of quantum observables of free fields is traditionally defined as the corresponding Wick algebra.
But the Wick algebra in quantum field theory may also be understood more systematically from first principles of quantization. It turns out that it is Moyal deformation quantization of the canonical Poisson bracket on the covariant phase space of the free field, which is the Peierls bracket modified to an almost Kähler structure by the 2-point function of a quasi-free Hadamard state (Dito 90, Dütsch-Fredenhagen 01). See example 2 and def. 7 below.
Understood in this form the construction directly generalized to quantum field theory on curved spacetimes (Brunetti-Fredenhagen 95, Brunetti-Fredenhagen 00, Hollands-Wald 01).
Finally, the shift by the quasi-free Hadamard state, which is the very source of the “normal ordering”, was understood as an example of the almost-Kähler version of the quantization recipe of Fedosov deformation quantization (Collini 16). For more on this see at locally covariant perturbative quantum field theory.
(almost Kähler vector space)
An almost Kähler vector space is a complex vector space $V$ equipped with two bilinear forms $\sigma, h \;\colon\; V \otimes_{\mathbb{R}} V \longrightarrow \mathbb{R}$ such that with $V$ regarded as a smooth manifold and with $\sigma, g$ regarded as constant tensors, then $(V,\sigma,h)$ is an almost Kähler manifold.
(standard almost Kähler vector spaces)
Let $V \coloneqq \mathbb{R}^2$ equipped with the complex structure given by the canonical identification $\mathbb{R}^2 \simeq \mathbb{C}$, let $\sigma \coloneqq \left( \array{0 & -1 \\ 1 & 0} \right)$ and $h \coloneqq \left( \array{ 1 & 0 \\ 0 & 1} \right)$. Then $(V,\sigma,g)$ is an almost Kähler vector space (def. 1).
(Wick algebra of an almost Kähler vector space)
Let $(\mathbb{R}^{2n},\sigma, g)$ be an almost Kähler vector space (def. 1). Then its Wick algebra is the formal power series vector space $\mathbb{C}[ [ a_1, a^\ast_1, \cdots, a_n, a^\ast_n ] ] [ [ \hbar ] ]$ equipped with the star product
given by the bilinear form
Here
is the ordinary (commutative) product in the formal power series algebra.
To make contact with the traditional notation we decorate the elements $P$ in the formal power series algebra with colons and declare the notation
(Wick algebra of a single mode)
Let $V \coloneqq \mathbb{R}^2 \simeq Span(\{x,y\})$ be a standard almost Kähler vector space according to example 1, with canonical coordinates denoted $x$ and $y$. We discuss its Wick algebra according to def. 2 and show that this reproduces the traditional definition of products of “normal ordered” operators as above.
To that end, consider the complex linear combination of the coordinates to the canonical complex coordinates $z \coloneqq x + i y$ and $\overline{z} \coloneqq x - i y$, which we suggestively write instead as
(with “$a$” the traditional symbol for the amplitude of a field mode).
We find the value of the almost-Kähler forms on these elements to be
Using this, we find the star product as follows (where we write $(-)\cdot (-)$ for the plain commutative product in the formal power series algebra):
These four cases are sufficient to see that in the star-product $P_1 \star_\omega P_2$ of general elements, we obtain correction term $\hbar$ to the ordinary commutative product precisely for every pair consisting of a factor of $a$ in $P_1$ and a factor $a^\ast$ in $P_2$. This is exactly the “normal ordering” prescription.
The genuine Wick algebras in quantum field theory are the analogues to those induced by finite-dimensional almost Kähler vector spaces as above, where now the underlying vector space is a space of (“microcausal”) functionals on the space of smooth functions on a spacetime, and where the symplectic form is given by the causal propagator of the wave equation/Klein-Gordon equation.
We give the definition below. First we briefly recall the relevant ingredients:
(point evaluation observable)
Let $X$ be a smooth manifold. Then the mapping space $C^\infty(X, \mathbb{R})$ is naturally a smooth space.
For $x \in X$ there is the smooth function
evaluating at $x \in X$.
More generally:
(smooth functionals on smooth functions)
For $f \in C_c^\infty(X)$ a bump function, it induces the function given by integration
where $dvol_g$ denotes the volume form of the given pseudo-Riemannian metric, and similarly if $f \in C_c^\infty(X^n)$ is a bump function on the $n$-fold Cartesian product manifold $X^n$, then there is
Write $\mathcal{F}_{reg}$ for the sub-algebra of smooth functions on the smooth space $C^\infty(X)$ which is generated from the functions $\Phi(f)$ for $n \in \mathbb{N}$ and $f \in C^\infty_c(X^n)$.
A further evident generalization of this takes $f \in \mathcal{E}'(X^n)$ to be a compactly supported distribution and induces the function
Write $\mathcal{F}_{dist}$ for the subalgebra generated by these functionals. (After deformation quantization below, this is the origin of “operator-valued distributions” in perturbative quantum field theory).
Recall the following general facts about the wave equation/Klein-Gordon equation:
Let $(X,g)$ be a time-oriented globally hyperbolic spacetime and let $m \in \mathbb{R}_{\geq 0}$ (the “mass”). Then the Klein-Gordon equation
(a partial differential equation on smooth functions $f \in C^\infty(X,\mathbb{R})$ ) has unique advanced and retarded Green functions $E^{R/A}$, namely continuous linear functionals
(from bump functions to general smooth functions) which are fundamental solutions in that
and which have advanced/retarded support of a distribution when viewed (via the Schwartz kernel theorem) as distributions on the Cartesian product manifold $X \times X$
In fact these two fundamental solutions are related by switching their arguments
Finally their wave front set is
Here the relation $(x_1, k_1) \sim (x_2, k_2)$ means that there exists a lightlike geodesic from $x_1$ to $x_2$ with cotangent vector $k_1$ at $x_1$ and $k_2$ at $x_2$.
It follows that the wave front set of their difference (the causal propagator)
is
This causal propagator gives the Poisson bracket on the covariant phase space of the free scalar field in that, as distributions
where on the left $\{-,-\}$ is the Poisson bracket and $\Phi(x)$ is the point evaluation function from def. 3. This means that on the algebra $\mathcal{F}_{reg}$ of regular functionals from def. 4 the bracket is given by
The point now is that:
The algebra $\mathcal{F}_{reg}$ of regular functionals (def. 4) is too small a subalgebra of all smooth functionals on $C^\infty(X)$ to be of interest (for instance the point interaction terms such as $g\Phi(x)^n \colon \phi \mapsto \int_X g(x) (\phi(x))^x dvol$ (for $g$ an adiabatic switching coupling constant) needed for phi^4 theory etc. are not contained). On the other hand $\mathcal{F}_{dist}$ is too large: extending the above Poisson bracket to this algebra would mean to form a product of distributions with $E$, and in general this is not defined.
But the obstruction to this product being defined is well characterized: One may multiply with $E$ all those distributions $f$ such that the sum of the wave front sets $WF(E) + WF(f)$ does not intersect the zero-section.
Therefore if $E$ could be modified such that its wave front set shrinks, then this variant may be applied to larger subalgebras of smooth functionals. Now def. 2 suggests that sensible modifications of the causal propagator are those by adding distributions $H$ on $X^2$ that are symmetric in their two arguments.
Such distributions are quasi-free Hadamard states:
Let $(X,g)$ be a time oriented globally hyperbolic spacetime.
A Hadamard 2-point function for the free scalar field on $(X,g)$ is a distribution
on the Cartesian product manifold such that
the anti-symmetric part of $\omega$ is the causal propagator $E$ (from prop. 1)
the wave front set is one causal half that of the causal propagator:
(Klein-Gordon bi-solution) $(\Box_g - m^2) \omega(-,x) = 0$ and $(\Box_g - m^2)\omega(x,-) = 0$, for all $x \in X$;
(positive semi-definiteness) For any complex-valued bump function $b$ we have that
(existence of Hadamard distributions)
Let $(X,g)$ be a globally hyperbolic spacetime. Then a Hadamard distribution $\omega$ according to def. 5 does exist.
Let $(X,g)$ be a globally hyperbolic spacetime.
Write $\mathcal{F}_{mc} \subset \mathcal{F}_{dist}$ for the subalgebra of smooth functionals
on the smooth space of smooth functions on $X$ which is generated from those distributions on some Cartesian product $X^n$ (as in def. 4) whose wave front set excludes those covectors to a point in $X^n$ all whose components are in the future cone or all whose components are in the past cone.
(After deformation quantization below, the distributions appearing in def. 6 are the origin of “operator-valued distributions” in perturbative quantum field theory).
(regular functionals are microcausal)
Every regular functional (def. 4) is a microcausal functional (def. 6), since the wave front set of a distribution that is given by an ordinary function is empty:
(adiabtaically switched point interactions are microcausal)
Let $g \in C^\infty_c(X)$ be a bump function, then for $n \in \mathbb{N}$ the smooth functional
is a microcausal functional (def. 6).
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.
For notational convenience, consider the case $n = 2$, the other cases are directly analogous. The distribution in question is the delta distribution
Now for $(x_1, x_2) \in X \times X$ and $\mathbb{R}^{2n} \simeq U \subset X \times X$ a chart around this point, the Fourier transform 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:
Since $g$ is a plain bump function, its Fourier transform $\hat g$ is quickly decaying (in the sense of wave front sets) with $k_1 + k_2$ (this prop.). Thus only on the cone $k_1 + k_2 = 0$ that function is in fact constant and in particular not decaying.
This means that the wave front set consists of the element of the form $(x, (k, -k))$ with $k \neq 0$. Since $k$ and $-k$ are both in the future cone or both in the 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:
(compactly supported polynomial local observables)
Write
for the space of differential forms on the jet bundle of $E$ which locally are polynomials in the field variables.
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 $\Sigma$ and polynomial with respect to the field variables (local observables).
Every $L \in \mathcal{F}_{loc}$ induces a functional
by integration of the pullback of $L$ along the jet prolongation of a given section:
These functionals happen to be microcausal, so that there is an inclusion
into the space of microcausal functionals. (See this example at microcausal functional).
(Hadamard-Moyal star product on microcausal functionals)
Let $(X,g)$ be a globally hyperbolic spacetime, and let $\omega \in \mathcal{D}'(X \times X)$ be a Hadamard distribution (def. 5) which is guaranteed to exist by prop. 2.
Then the star product
on microcausal functionals $P_1, P_2 \in \mathcal{F}_{mc}$ is well defined in that the products of distributions that appear in expanding out the exponential are such that the sum of the wave front sets of the factors does not intersect the zero section.
By definition of Hadamard distribution, the wave front set of powers of $\omega$ has all cotangents on the first variables future pointing, and all those on the second variables past pointing. The first variables are integrated against those of $P_1$ and the second against $P_2$. By definition of microcausal functionals, the wave front sets of $P_1$ and $P_2$ are disjoint from the subsets where all components are future pointing or all components are past-pointing. Therefore the relevant sum of of the wave front covectors never vanishes.
(Wick algebra of free quantum field)
Let $(X,g)$ be a globally hyperbolic spacetime and let $\omega \in \mathcal{D}'(X \times X)$ be a Hadamard distribution (def. 5) which is guaranteed to exist by prop. 2.
Then the Wick algebra of quantum observables of the free scalar field on $(X,g)$ is the space of microcausal functionals $\mathcal{F}_{mc}$ (def. 6) equipped with the Hadamard-Moyal star product from prop. 3:
need to quotient out ideal of elements in the image of $\Box_g - m^2$ to go on shell
In Minkowski spacetime the Hadamard state is simply the usual vacuum state $\vert vac \rangle$, hence the Hadamard distribution is, as a generalized function
Green functions for the Klein-Gordon operator on a globally hyperbolic spacetime:
propagator | $\phantom{AA}$ | $\phantom{AA}$ primed wave front set | on Minkowski spacetime | generally |
---|---|---|---|---|
causal propagator | $\array{\Delta \coloneqq \Delta_R - \Delta_A }$ | $\array{- \\ \phantom{A} \\ \phantom{a}}$ | $\array{\Delta_S(x,y) = \\ \langle vac\vert [\Phi(x),\Phi(y)] \vert vac\rangle }$ | Peierls-Poisson bracket |
advanced propagator | $\Delta_A$ | $\array{\Delta_A(x,y) = \\ \Theta((y-x)^0)\langle vac\vert [\Phi(x),\Phi(y)] \vert vac\rangle }$ | ||
retarded propagator | $\Delta_R$ | $\array{\Delta_R(x,y) = \\ \Theta((x-y)^0)\langle vac\vert [\Phi(x),\Phi(y)] \vert vac\rangle }$ | ||
Dirac propagator | $\Delta_D = \tfrac{1}{2}(\Delta_A + \Delta_R)$ | $\array{+ \\ \phantom{A} \\ \phantom{a}}$ | ||
Hadamard propagator | $\begin{aligned} \omega &= \tfrac{i}{2}\Delta + H \\ & = \omega_F - i \Delta_A \end{aligned}$ | $\array{\omega(x,y) = \\ \langle vac \vert \Phi(x) \Phi(y) \vert vac \rangle }$ | normal-ordered product (2-point function of quasi-free state) | |
Feynman propagator | $\array{\omega_F & = i \Delta_D + H \\ & = \omega + i \Delta_A}$ | $\array{E_F(x,y) = \\ \langle vac \vert T(\Phi(x)\Phi(y)) \vert vac \rangle }$ | time-ordered product |
(see also Kocic’s overview: pdf)
Therefore the abstractly defined Wick algebra as in def. 7 in this case satisfies the relation
This is the traditional expression for the normal ordered Wick product on Minkowski spacetime (e.g. here).
The construction goes back to
Its realization as the Moyal deformation quantization of the Peierls bracket shifted by a quasi-free Hadamard state is due to
further amplified in
and the generalization to quantum field theory on curved spacetime is discussed in
Romeo Brunetti, Klaus Fredenhagen, M. Köhler, The microlocal spectrum condition and Wick polynomials on curved spacetimes, Commun. Math. Phys. 180, 633-652, 1996 (arXiv:gr-qc/9510056)
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)
Stefan Hollands, Robert Wald, Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime, Commun. Math. Phys. 223:289-326, 2001 (arXiv:gr-qc/0103074)
The conceptual explanation of the shift by the Hadamard state as a step in the almost-Kähler version of Fedosov deformation quantization is due to