symmetric monoidal (∞,1)-category of spectra
In deformation quantization of Poisson manifolds the commutative product $\cdot$ of the commutative algebra of functions is replaced by a noncommutative associative product. This is often called a star product and denoted “$\star$”.
An archetypical example is the Moyal star product (example 3 below) that deforms the function algebra on a Poisson vector space, and often “star product” is by default understood to be a Moyal star product. Indeed every star product induced from a constant rank-2 tensor on a vector space is isomorphic to a Moyal star product (prop. 2 below).
More recently also nonassociative “star products” have been proposed to be of interest.
Let $V$ be a vector space of finite dimension and let $\omega \in V \otimes V$ be an element of the tensor product (not necessarily skew symmetric at the moment).
We may canonically regard $V$ as a smooth manifold, in which case $\omega$ is canonically regarded as a constant rank-2 tensor. As such it has a canonical action by forming derivatives on the tensor product of the space of smooth functions:
If $\{\partial_i\}$ is a linear basis for $V$, identified, as before, with a basis for $\Gamma(T V)$, then in this basis this operation reads
where $\partial_i f \coloneqq \frac{\partial f}{\partial x^i}$ denotes the partial derivative of the smooth function $f$ along the $i$th coordinate, and where we use the Einstein summation convention.
For emphasis we write
for the pointwise product of smooth functions.
(star product induced by constant rank-2 tensor)
Given $(V,\omega)$ as above, then the star product induced by $\omega$ on the formal power series algebra $C^\infty(V) [ [\hbar] ]$ in a formal variable $\hbar$ (“Planck's constant”) with coefficients in the smooth functions on $V$ is the linear map
given by
Hence
(star product is associative and unital)
Given $(V,\omega)$ as above, then the star product $(-) \star_\omega (-)$ from def. 1 is associative and unital with unit the constant function $1 \in C^\infty(V) \hookrightarrow C^\infty(V)[ [ \hbar ] ]$.
Hence the vector space $C^\infty(V)$ equipped with the star product $\omega$ is a unital associative algebra.
Observe that the product rule of differentiation says that
Using this we compute as follows:
In the last line we used that the ordinary pointwise product of functions is associative, and wrote $prod_3 \colon C^\infty(V) \otimes C^\infty(V) \otimes C^\infty(V) \to C^\infty(V)$ for the unique pointwise product of three functions.
The last expression above is manifestly independent of the choice of order of the arguments in the triple star product, and hence it is clear that an analogous computation yields
(shift by symmetric contribution is isomorphism of star product)
Let $V$ be a vector space, $\omega \in V \otimes V$ a rank-2 tensor and $\alpha \in Sym(V \otimes V)$ a symmetric rank-2 tensor.
Then the linear map
constitutes an isomorphism of star product algebras (prop. 1) of the form
hence identifying the star product induced from $\omega$ with that induced from $\omega + \alpha$.
In particular every star product algebra $(C^\infty(V)[ [\hbar] ],\star_\omega)$ is isomorphic to a Moyal star product algebra $\star_{\tfrac{1}{2}\pi}$ (example 3) with $\tfrac{1}{2}\pi^{i j} \tfrac{1}{2}(\omega^{i j} - \omega^{j i})$ the skew-symmetric part of $\omega$, this isomorphism being exhibited by $\alpha^{i j} = - \tfrac{1}{2}(\omega^{i j} + \omega^{j i})$ (minus) the symmetric part.
We need to show that
To this end, observe that the product rule of differentiation applied twice in a row implies that
Using this we compute
Some examples of star products as in def. 1:
If $\omega = 0$ in def. 1, then the star product $\star_0 = \cdot$ is the plain pointwise product.
If $\omega = \tfrac{1}{2}\pi$ in def. 1 is skew-symmetric, it may be regarded as a constant Poisson tensor $\pi$ on the smooth manifold $V$. In this case $\star_{\tfrac{1}{2}\pi}$ is called a Moyal star product and the star-product algebra $C^\infty(V)[ [\hbar] ], \star_\pi)$ is called the Moyal deformation quantization of the Poisson manifold $(V,\pi)$.
(normal-ordered products and time-ordered products)
In cases where $V$ is not finite dimensional, but for instance a space of sections of a linear field bundle, the expression involved in the definition of the star product in def. 1 still may be interpreted as products of distributions, but these only exist if certain wavefront set-conditions are met.
In these situations it happens that the star product for some Poisson tensor $\tfrac{1}{2}\pi$ is not defined due to wavefront set collision, but the star product $\star_{\tfrac{1}{2}\pi + \alpha}$ for the tensor $\tfrac{1}{2}\pi + \alpha$ shifted by a symmetric contribition $\alpha$ as in prop. 2 is defined, and serves as the proper stand-in for the non-existing $\star_{\tfrac{1}{2}\pi}$.
This is the case notably for Wick algebras (algebras of quantum observables of free fields): here $\tfrac{1}{2}\pi$ is the causal propagator for which $\star_\omega$ does not exist, and $\omega = \tfrac{1}{2}\pi + \alpha$ is a Hadamard propagator, so that $\star_{\omega}$ does exist on microcausal functionals: it is the “normal-ordered product” of quantum observables.