star product



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 VV be a vector space of finite dimension and let ωVV\omega \in V \otimes V be an element of the tensor product (not necessarily skew symmetric at the moment).

We may canonically regard VV as a smooth manifold, in which case ω\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:

ω:C (V)C (V)C (V)C (V). \omega \;\colon\; C^\infty(V) \otimes C^\infty(V) \longrightarrow C^\infty(V) \otimes C^\infty(V) \,.

If { i}\{\partial_i\} is a linear basis for VV, identified, as before, with a basis for Γ(TV)\Gamma(T V), then in this basis this operation reads

ω(fg)=ω ij( if)( jg), \omega(f \otimes g) \;=\; \omega^{i j} (\partial_i f) \otimes (\partial_j g) \,,

where iffx i\partial_i f \coloneqq \frac{\partial f}{\partial x^i} denotes the partial derivative of the smooth function ff along the iith coordinate, and where we use the Einstein summation convention.

For emphasis we write

C (V)C (V) prod C (V) fg fg \array{ C^\infty(V) \otimes C^\infty(V) &\overset{prod}{\longrightarrow}& C^\infty(V) \\ f \otimes g &\mapsto& f \cdot g }

for the pointwise product of smooth functions.


(star product induced by constant rank-2 tensor)

Given (V,ω)(V,\omega) as above, then the star product induced by ω\omega on the formal power series algebra C (V)[[]]C^\infty(V) [ [\hbar] ] in a formal variable \hbar (“Planck's constant”) with coefficients in the smooth functions on VV is the linear map

() ω():C (V)[[]]C (V)[[]]C (V)[[]] (-) \star_\omega (-) \;\colon\; C^\infty(V)[ [ \hbar ] ] \otimes C^\infty(V)[ [ \hbar ] ] \longrightarrow C^\infty(V)[ [\hbar] ]

given by

() ω()prodexp(ω ij i j) (-) \star_\omega (-) \;\coloneqq\; prod \circ \exp\left( \hbar \omega^{i j} \partial_i \otimes \partial_j \right)


f ωg1+ω ijfx igx j+ 212ω ijω kl 2fx ix k 2gx jx l+. f \star_\omega g \;\coloneqq\; 1 + \hbar \omega^{i j} \frac{\partial f}{\partial x^i} \cdot \frac{\partial g}{\partial x^j} + \hbar^2 \tfrac{ 1 }{2} \omega^{i j} \omega^{k l} \frac{\partial^2 f}{\partial x^{i} \partial x^{k}} \cdot \frac{\partial^2 g}{\partial x^{j} \partial x^{l}} + \cdots \,.

(star product is associative and unital)

Given (V,ω)(V,\omega) as above, then the star product () ω()(-) \star_\omega (-) from def. 1 is associative and unital with unit the constant function 1C (V)C (V)[[]]1 \in C^\infty(V) \hookrightarrow C^\infty(V)[ [ \hbar ] ].

Hence the vector space C (V)C^\infty(V) equipped with the star product ω\omega is a unital associative algebra.


Observe that the product rule of differentiation says that

iprod=prod( iid+id i). \partial_i \circ prod = prod \circ ( \partial_i \otimes id \;+\; id \otimes \partial_i ) \,.

Using this we compute as follows:

(f ωg) ωh =prodexp(ω ij i j)((prodexp(ω kl k l))id)(fgg) =prodexp(ω ij i j)(prodid)(exp(ω kl k l)id)(fgg) =prod(prodid)exp(ω ij( iid j+id i j)exp(ω kl k l)id(fgg) =prod(prodid)exp(ω ij iid j)exp(ω ijid i j)exp(ω kl k lid)(fgg) =prod 3exp(ω ij( i jid+ iid j+id i j)) \begin{aligned} (f \star_\omega g) \star_\omega h & = prod \circ \exp( \omega^{i j} \partial_i \otimes \partial_j ) \circ \left( \left( prod \circ \exp( \omega^{k l} \partial_k \otimes \partial_l ) \right) \otimes id \right) (f \otimes g \otimes g) \\ & = prod \circ \exp( \omega^{i j} \partial_i \otimes \partial_j ) \circ (prod \otimes id) \circ \left( \exp( \omega^{k l} \partial_k \otimes \partial_l ) \otimes id \right) (f \otimes g \otimes g) \\ & = prod \circ (prod \otimes id) \circ \exp( \omega^{i j} ( \partial_i \otimes id \otimes \partial_j +id \otimes \partial_i \otimes \partial_j ) \circ \exp( \omega^{k l} \partial_k \otimes \partial_l ) \otimes id (f \otimes g \otimes g) \\ & = prod \circ (prod \otimes id) \circ \exp( \omega^{i j} \partial_i \otimes id \otimes \partial_j ) \circ \exp( \omega^{i j} id \otimes \partial_i \otimes \partial_j ) \circ \exp( \omega^{k l} \partial_k \otimes \partial_l \otimes id ) (f \otimes g \otimes g) \\ & = prod_3 \circ \exp( \omega^{i j} ( \partial_i \otimes \partial_j \otimes id + \partial_i \otimes id \otimes \partial_j + id \otimes \partial_i \otimes \partial_j) ) \end{aligned}

In the last line we used that the ordinary pointwise product of functions is associative, and wrote prod 3:C (V)C (V)C (V)C (V)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

=f ω(g ωh). \cdots = f \star_\omega (g \star_\omega h) \,.



(shift by symmetric contribution is isomorphism of star product)

Let VV be a vector space, ωVV\omega \in V \otimes V a rank-2 tensor and αSym(VV)\alpha \in Sym(V \otimes V) a symmetric rank-2 tensor.

Then the linear map

C (V) exp(12α) C (V) f exp(12α ij i j)f \array{ C^\infty(V) &\overset{\exp\left(-\hbar\tfrac{1}{2}\alpha \right)}{\longrightarrow}& C^\infty(V) \\ f &\mapsto& \exp\left( -\hbar\tfrac{1}{2}\alpha^{i j} \partial_i \partial_j \right) f }

constitutes an isomorphism of star product algebras (prop. 1) of the form

exp(12α):(C (V)[[]], ω)(C (V))[[]], ω+α), \exp\left(-\hbar\tfrac{1}{2}\alpha \right) \;\colon\; (C^\infty(V)[ [\hbar] ], \star_\omega) \overset{\simeq}{\longrightarrow} (C^\infty(V))[ [\hbar] ], \star_{\omega + \alpha}) \,,

hence identifying the star product induced from ω\omega with that induced from ω+α\omega + \alpha.

In particular every star product algebra (C (V)[[]], ω)(C^\infty(V)[ [\hbar] ],\star_\omega) is isomorphic to a Moyal star product algebra 12π\star_{\tfrac{1}{2}\pi} (example 3) with 12π ij12(ω ijω ji)\tfrac{1}{2}\pi^{i j} \tfrac{1}{2}(\omega^{i j} - \omega^{j i}) the skew-symmetric part of ω\omega, this isomorphism being exhibited by α ij=12(ω ij+ω ji)\alpha^{i j} = - \tfrac{1}{2}(\omega^{i j} + \omega^{j i}) (minus) the symmetric part.


We need to show that

prodexp(ω)(exp(12α)exp(12α))=exp(12α)prodexp(ω). prod \circ \exp( \omega ) \circ \left( \exp\left( -\tfrac{1}{2}\alpha\right) \otimes \exp\left( -\tfrac{1}{2}\alpha \right) \right) \;=\; \exp\left( -\tfrac{1}{2}\alpha \right) \circ prod \circ \exp( \omega ) \,.

To this end, observe that the product rule of differentiation applied twice in a row implies that

i jprod=prod(( i j)id+id( i j)+ i j+ j i). \partial_i \partial_j \circ prod \;=\; prod \circ \left( (\partial_i \partial_j) \otimes id + id \otimes (\partial_i \partial_j) + \partial_i \otimes \partial_j + \partial_j \otimes \partial_i \right) \,.

Using this we compute

exp(12α ij i j)prodexp(ω ij k l) =prodexp(12α ij( i j)id+id( i j)+ i j+ j i)exp(ω ij k l) =prodexp((ω ij+α ij) i j)exp(12α ij( i j)id12α ijid( i j)) =prodexp((ω ij+α ij) i j)(exp(12α)exp(12α)) \begin{aligned} \exp\left( - \hbar\tfrac{1}{2}\alpha^{i j} \partial_i \partial_j \right) \circ prod \circ \exp( \hbar \omega^{i j} \partial_{k} \otimes \partial_l ) & = prod \circ \exp\left( - \hbar \tfrac{1}{2}\alpha^{i j} (\partial_i \partial_j) \otimes id + id \otimes (\partial_i \partial_j) + \partial_i \otimes \partial_j + \partial_j \otimes \partial_i \right) \circ \exp( \hbar \omega^{i j} \partial_{k} \otimes \partial_l ) \\ & = prod \circ \exp\left( \hbar (\omega^{i j} + \alpha^{i j}) \partial_i \otimes \partial_j \right) \circ \exp\left( - \hbar \tfrac{1}{2} \alpha^{i j} (\partial_i \partial_j) \otimes id - \hbar \tfrac{1}{2} \alpha^{i j} id \otimes (\partial_i \partial_j) \right) \\ & = prod \circ \exp\left( \hbar (\omega^{i j} + \alpha^{i j}) \partial_i \otimes \partial_j \right) \circ \left( \exp(-\hbar \tfrac{1}{2}\alpha) \otimes \exp(-\hbar \tfrac{1}{2}\alpha) \right) \end{aligned}


Some examples of star products as in def. 1:


If ω=0\omega = 0 in def. 1, then the star product 0=\star_0 = \cdot is the plain pointwise product.


(Moyal star product)

If ω=12π\omega = \tfrac{1}{2}\pi in def. 1 is skew-symmetric, it may be regarded as a constant Poisson tensor π\pi on the smooth manifold VV. In this case 12π\star_{\tfrac{1}{2}\pi} is called a Moyal star product and the star-product algebra C (V)[[]], π)C^\infty(V)[ [\hbar] ], \star_\pi) is called the Moyal deformation quantization of the Poisson manifold (V,π)(V,\pi).


(normal-ordered products and time-ordered products)

In cases where VV 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 12π\tfrac{1}{2}\pi is not defined due to wavefront set collision, but the star product 12π+α\star_{\tfrac{1}{2}\pi + \alpha} for the tensor 12π+α\tfrac{1}{2}\pi + \alpha shifted by a symmetric contribition α\alpha as in prop. 2 is defined, and serves as the proper stand-in for the non-existing 12π\star_{\tfrac{1}{2}\pi}.

This is the case notably for Wick algebras (algebras of quantum observables of free fields): here 12π\tfrac{1}{2}\pi is the causal propagator for which ω\star_\omega does not exist, and ω=12π+α\omega = \tfrac{1}{2}\pi + \alpha is a Hadamard propagator, so that ω\star_{\omega} does exist on microcausal functionals: it is the “normal-ordered product” of quantum observables.

Revised on September 28, 2017 23:37:37 by David Roberts (