star product

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+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
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[[!include higher algebra - contents]]
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#### Geometric quantization
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[[!include geometric quantization - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

In [[deformation quantization]] of [[Poisson manifolds]] the commutative product $\cdot$ of the [[commutative algebra|commutative]] [[algebra of functions]] is replaced by a noncommutative associative product. This is often called a **star product** and denoted "$\star$".

An [[archetypal example]] is the _[[Moyal star product]]_ (example \ref{MoyalStarProduct} 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 [[isomorphism|isomorphic]] to a Moyal star product (prop. \ref{SymmetricContribution} below).

More recently also nonassociative "star products" have been proposed to be of interest.

## Definition

### On finite-dimensional vector spaces

Let $V$ be a [[finite dimensional vector space]] and let $\pi \in V \otimes V$ 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 $\pi$ 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]]:

$$
  \pi
  \;\colon\;
  C^\infty(V) \otimes C^\infty(V)
  \longrightarrow
  C^\infty(V) \otimes C^\infty(V)
  \,.
$$

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

$$
  \pi(f \otimes g)
  \;=\;
  \pi^{i j}
  (\partial_i f) \otimes (\partial_j g)
  \,,
$$

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

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

for the pointwise product of smooth functions.

+-- {: .num_defn #StarPoduct}
###### Definition
**([[star product]] induced by constant rank-2 [[tensor]])**

Given $(V,\pi)$ as above, then the _[[star product]]_ induced by $\pi$ 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

$$
  (-) \star_\pi (-)
  \;\colon\;
  C^\infty(V)[ [ \hbar ] ] \otimes C^\infty(V)[ [ \hbar ] ]
    \longrightarrow
  C^\infty(V)[ [\hbar] ]
$$

given by

$$
  (-) \star_\pi (-)
  \;\coloneqq\;
  prod
    \circ \exp\left(
       \hbar \pi^{i j} \frac{\partial}{\partial x^i} \otimes \frac{\partial}{\partial x^j}
  \right)
$$

Hence

$$
  f \star_\pi g
  \;\coloneqq\;
  1
  +
  \hbar \pi^{i j}
  \frac{\partial f}{\partial x^i} \cdot \frac{\partial g}{\partial x^j}
  +
  \hbar^2
  \tfrac{ 1 }{2}
  \pi^{i j}
  \pi^{k l}
  \frac{\partial^2 f}{\partial x^{i} \partial x^{k}}
    \cdot
  \frac{\partial^2 g}{\partial x^{j} \partial x^{l}}
  +
  \cdots
  \,.
$$

=--

+-- {: .num_prop #AssociativeAndUnitalStarProduct}
###### Proposition
**([[star product]] is [[associativity|associative]] and [[unitality|unital]])**

Given $(V,\pi)$ as above, then the star product
$(-) \star_\pi (-)$ from def. \ref{StarPoduct}
is [[associativity|associative]] and [[unitality|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]] $\pi$ is a [[unital algebra|unital]] [[associative algebra]].

=--

+-- {: .proof}
###### Proof

Observe that the [[product rule]] of [[differentiation]] says that

$$
  \partial_i \circ prod
    =
   prod \circ ( \partial_i \otimes id \;+\; id \otimes \partial_i  )
  \,.
$$

Using this we compute as follows:

$$
  \begin{aligned}
  (f \star_\pi  g) \star_\pi h
  & =
  prod \circ
  \exp( \pi^{i j} \partial_i \otimes \partial_j )  \circ
  \left(
    \left( prod \circ \exp( \pi^{k l} \partial_k \otimes \partial_l ) \right) \otimes id
  \right)
  (f \otimes g \otimes g)
  \\
  & =
  prod \circ
  \exp( \pi^{i j} \partial_i \otimes \partial_j ) \circ
  (prod \otimes id)
  \circ
  \left(
    \exp( \pi^{k l} \partial_k \otimes \partial_l )  \otimes id
  \right)
  (f \otimes g \otimes g)
  \\
  & =
  prod \circ
  (prod \otimes id) \circ
  \exp( \pi^{i j}  ( \partial_i \otimes id \otimes \partial_j +id \otimes \partial_i \otimes \partial_j   )
  \circ
  \exp( \pi^{k l} \partial_k \otimes \partial_l )  \otimes id
  (f \otimes g \otimes g)
  \\
  & =
  prod \circ
  (prod \otimes id) \circ
  \exp( \pi^{i j}   \partial_i \otimes id \otimes \partial_j   )
  \circ
  \exp( \pi^{i j}  id \otimes \partial_i \otimes \partial_j )
  \circ
  \exp( \pi^{k l} \partial_k \otimes \partial_l \otimes id )
  (f \otimes g \otimes g)
  \\
  & =
  prod_3
  \circ
  \exp( \pi^{i j} ( \partial_i \otimes \partial_j \otimes id  +  \partial_i \otimes id \otimes \partial_j + id \otimes \partial_i \otimes \partial_j)   )
  \end{aligned}
$$

In the last line we used that the ordinary pointwise product of functions is associative, and wrote $prod_3 \colon C^\infty(V) \otimes C^\infty(V) \otimes C^\infty(V) \to C^\infty(V)$ 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

$$
  \cdots = f \star_\pi (g \star_\pi h)
  \,.
$$

=--

### On polynomial observables in field theory

+-- {: .num_defn #PropagatorStarProduct}
###### Definition
**([[star products]] on [[regular polynomial observables]] induced from [[propagators]])**

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] with [[field bundle]] $E \overset{fb}{\to} \Sigma$, and let $\Delta \in \Gamma'_\Sigma((E \boxtimes E)^\ast)$ be a [[distribution of two variables]] on [[field histories]].

On the [[off-shell]] [[regular polynomial observables]] with a [[formal power series|formal paramater]] $\hbar$ adjoined consider the bilinear map

$$
  PolyObs(E)_{reg}[ [\hbar] ]
    \otimes
  PolyObs(E)_{reg} [ [ \hbar ] ]
    \overset{\star_{\Delta}}{\longrightarrow}
  PolyObs(E)_{reg}[ [\hbar] ]
$$

given as in def. \ref{StarPoduct}, but with [[partial derivatives]] replaced by [[functional derivatives]]

$$
  A_1 \star_{\Delta} A_2
  \;\coloneqq\;
  ((-)\cdot(-))
  \circ
  \exp\left(
    \int_\Sigma \Delta^{a b}(x,y)
    \frac{\delta}{\delta \mathbf{\Phi}^a(x)}
      \otimes
    \frac{\delta}{\delta \mathbf{\Phi}^b(y)}
  \right)
  (A_1 \otimes A_2)
$$

As in prop. \ref{AssociativeAndUnitalStarProduct} this defines a [[unital algebra|unital]] and [[associative algebra]] [[structure]].

If the [[Euler-Lagrange equations|Euler-Lagrange]] [[equations of motion]] induced by the [[Lagrangian density]] $\mathbf{L}$ are [[Green hyperbolic differential equations]] and if $\Delta$ is a [[propagator]] for these [[differential equations]], then this [[star product]] algebra descends to the [[on-shell]] [[regular polynomial observables]]

$$
  PolyObs(E,\mathbf{L})_{reg}[ [\hbar] ]
    \otimes
  PolyObs(E, \mathbf{L})_{reg} [ [ \hbar ] ]
    \overset{\star_{\Delta}}{\longrightarrow}
  PolyObs(E, \mathbf{L})_{reg}[ [\hbar] ]
  \,.
$$

=--

([Dito 90](deformation+quantization#Dito90), [Dütsch-Fredenhagen 00](deformation+quantization#DuetschFredenhagen00) [Dütsch-Fredenhagen 01](deformation+quantization#DuetschFredenhagen01), [Hirshfeld-Henselder 02](deformation+quantization#HirschfeldHenselder02))

## Properties

### Equivalences of star products

+-- {: .num_prop #SymmetricContribution}
###### Proposition
**(shift by symmetric contribution is [[isomorphism]] of [[star products]])**

Let $V$ be a vector space, $\pi \in V \otimes V$ a rank-2 [[tensor]] and $\alpha \in Sym(V \otimes V)$ a _symmetric_ rank-2 tensor.

Then the [[linear map]]

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

constitutes an [[isomorphism]] of star product algebras (prop. \ref{AssociativeAndUnitalStarProduct}) of the form

$$
  \exp\left(\hbar\tfrac{1}{2}\hbar\alpha \right)
   \;\colon\;
  (C^\infty(V)[ [\hbar] ], \star_{\pi})
    \overset{\simeq}{\longrightarrow}
  (C^\infty(V))[ [\hbar] ], \star_{\pi + \alpha})
  \,,
$$

hence identifying the star product induced from $\pi$ with that induced from $\pi + \alpha$.

In particular every star product algebra $(C^\infty(V)[ [\hbar] ],\star_\pi)$ is isomorphic to a Moyal star product algebra $\star_{\tfrac{1}{2}\pi}$ (example \ref{MoyalStarProduct}) with $\tfrac{1}{2}\pi_{skew}^{i j} = \tfrac{1}{2}(\pi^{i j} - \pi^{j i})$ the skew-symmetric part of $\pi$, this isomorphism being exhibited by the symmetric part $2\alpha^{i j} = \tfrac{1}{2}(\pi^{i j} + \pi^{j i})$.

=--

+-- {: .proof}
###### Proof

We need to show that

$$
  \array{
    C^\infty(V)[ [\hbar] ] \otimes  C^\infty(V)[ [\hbar] ]
    &
     \overset{
       \exp\left( \tfrac{1}{2}\hbar \alpha \right)
       \otimes
       \exp\left( \tfrac{1}{2}\hbar \alpha \right)
     }{\longrightarrow}&
    C^\infty(V)[ [\hbar] ] \otimes  C^\infty(V)[ [\hbar] ]
     \\
     {}^{\mathllap{\star_{\pi}}}\downarrow
       &&
    \downarrow^{\mathrlap{\star_{\pi + \alpha}}}
    \\
    C^\infty(V)[ [\hbar] ]
      &\underset{\exp\left( \tfrac{1}{2} \alpha \right) }{\longrightarrow}&
    C^\infty(V)[ [\hbar] ]
  }
$$

hence that

$$
  prod \circ
  \exp( \hbar(\pi + \alpha) )
  \circ
  \left(
    \exp\left( \tfrac{1}{2}\alpha\right)
    \otimes
    \exp\left( \tfrac{1}{2}\alpha \right)
  \right)
  \;=\;
  \exp\left( \tfrac{1}{2}\alpha \right)
  \circ
   prod \circ \exp( \pi )
  \,.
$$

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

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

Using this we compute

$$
  \begin{aligned}
    \exp\left( \hbar\tfrac{1}{2}\alpha^{i j} \partial_i \partial_j \right)
    \circ
    prod
    \circ
    \exp( \hbar \pi^{i j} \partial_{i} \otimes \partial_j )
    & =
    prod
      \circ
      \exp\left(
        \hbar \tfrac{1}{2}\alpha^{i j}
        \left(
           (\partial_i \partial_j) \otimes id
           +
           id \otimes (\partial_i \partial_j)
           +
           \partial_i \otimes \partial_j
           +
           \partial_j \otimes \partial_i
        \right)
      \right)
      \circ
    \exp( \hbar \pi^{i j} \partial_{k} \otimes \partial_l )
    \\
    &
    =
    prod
    \circ
    \exp\left(
       \hbar (\pi^{i j} + \alpha^{i j}) \partial_i \otimes \partial_j
    \right)
    \circ
    \exp\left(
      \hbar \tfrac{1}{2} \alpha^{i j} (\partial_i \partial_j) \otimes id
      \hbar \tfrac{1}{2} \alpha^{i j} id \otimes (\partial_i \partial_j)
    \right)
    \\
    & =
    prod
    \circ
    \exp\left(
       \hbar (\pi^{i j} + \alpha^{i j}) \partial_i \otimes \partial_j
    \right)
    \circ
    \left(
      \exp\left( \tfrac{1}{2} \hbar \alpha \right)
        \otimes
      \exp\left( \tfrac{1}{2} \hbar \alpha \right)
    \right)
  \end{aligned}
$$

=--

### Integral representations

+-- {: .num_prop #IntegralRepresentationOfStarProduct}
###### Proposition
**([[integral]] representation of [[star product]])**

If $\pi$ skew-symmetric and invertible, in that there exists $\omega \in V^\ast \otimes V^\ast$ with $\pi^{i j}\omega_{j k} = \delta^i_k$, and if the functions $f,g$ admit [[Fourier analysis]] (are [[functions with rapidly decreasing partial derivatives]]), then  their [[star product]] (def. \ref{StarPoduct}) is equivalently given by the following [[integral]] expression:

$$
  \begin{aligned}
  \left(f \star_\pi g\right)(x)
   &=
  \frac{(det(\omega)^{2n})}{(2 \pi \hbar)^{2n} }
  \int
    e^{ \tfrac{1}{i \hbar} \omega((x - \tilde y),(x-y))}
    f(y)
    g(\tilde y)
   \,
   d^{2 n} y
   \,
   d^{2 n} \tilde y
  \end{aligned}
$$

=--

([Baker 58](Moyal+deformation+quantization#Baker58))

+-- {: .proof}
###### Proof

We compute as follows:

$$
  \begin{aligned}
  \left(f \star_\pi g\right)(x)
  & \coloneqq
  prod
    \circ \exp\left(
       \hbar \pi^{i j} \frac{\partial}{\partial x^i} \otimes \frac{\partial}{\partial x^j}
  \right)(f, g)
  \\
  & =
  \frac{1}{(2 \pi)^{2n}}
  \frac{1}{(2 \pi)^{2n}}
  \int
  \int
    \underbrace{
      e^{ i \hbar \pi(k,q) }
    }
    \underbrace{
      e^{i k \cdot (x-y)}
      f(y)
    }
    \underbrace{
      e^{i q \cdot (x- \tilde y)}
      g(\tilde y)
    }
   \,
   d^{2 n} k
   \,
   d^{2 n} q
   \,
   d^{2 n} y
   \,
   d^{2 n} \tilde y
   \\
   & =
  \frac{1}{(2 \pi)^{2n}}
  \int
    \delta\left( x - \tilde y + \hbar \pi \cdot k \right)
    e^{i k \cdot (x-y)}
    f(y)
    g(\tilde y)
   \,
   d^{2 n} k
   \,
   d^{2 n} y
   \,
   d^{2 n} \tilde y
   \\
   & =
  \frac{1}{(2 \pi)^{2n}}
  \int
    \delta\left( x - \tilde y + z \right)
    e^{ \tfrac{i}{\hbar} \omega(z, (x-y))}
    f(y)
    g(\tilde y)
   \,
   d^{2 n} z
   \,
   d^{2 n} y
   \,
   d^{2 n} \tilde y
  \\
  & =
  \frac{(det(\pi)^{2n})}{(2 \pi \hbar)^{2n} }
  \int
    e^{\tfrac{1}{i \hbar}\omega((x - \tilde y),(x-y))}
    f(y)
    g(\tilde y)
   \,
   d^{2 n} y
   \,
   d^{2 n} \tilde y
  \end{aligned}
$$

Here in the first step we expressed $f$ and $g$ both by their [[Fourier transform]] (inserting the Fourier expression of the [[delta distribution]] from [this example](Dirac+distribution#FourierTransformOfDeltaDistribution)) and used that under this transformation the [[partial derivative]] $\pi^{a b} \frac{\partial}{\partial\phi^a}{\frac{\partial}{\phi^b}}$ turns into the product with $i \pi^{i j} k_i k_j$ ([this prop.](Fourier+transform#BasicPropertiesOfFourierTransformOverCartesianSpaces)).
Then we identified again the Fourier-expansion of a [[delta distribution]]
and finally we applied the [[change of integration variables]] $k = \tfrac{1}{\hbar}\omega \cdot z$ and then evaluated the [[delta distribution]].

=--

Next we express this as the [[groupoid convolution product]] of polarized sections of the [[symplectic groupoid]]. To this end, we first need the following definnition:

+-- {: .num_defn #SymplecticGroupoidOfSymplecticVectorSpace}
###### Definition
**([[symplectic groupoid]] of [[symplectic vector space]])**

Assume that $\pi$ is the inverse of a [[symplectic form]] $\omega$ on $\mathbb{R}^{2n}$. Then the [[Cartesian product]]

$$
  \array{
    && \mathbb{R}^{2n} \times \mathbb{R}^{2n}
    \\
    & {}^{\mathllap{pr_1}}\swarrow && \searrow^{\mathrlap{pr_2}}
    \\
    \mathbb{R}^{2n} && && \mathbb{R}^{2n}
  }
$$

inherits the symplectic structure

$$
  \Omega
  \;\coloneqq\;
  \left(
    pr_1^\ast \omega - pr_2^\ast \omega
  \right)
$$

given by

$$
  \begin{aligned}
    \Omega
    & =
    \omega_{i j} d x^i \wedge d x^j - \omega_{i j} d y^i \wedge d y^j
    \\
    & =
    \omega_{i j} ( d x^i - d y^i ) \wedge ( d x^j +  d y^j )
  \end{aligned}
  \,.
$$

The [[pair groupoid]] on $\mathbb{R}^{2n}$ equipped with this [[symplectic form]] on its space of [[morphisms]] is a [[symplectic groupoid]].

A choice of potential form $\Theta$ for $\Omega$, hence with $\Omega = d \Theta$, is given by

$$
  \Theta
   \coloneqq
   -\omega_{i j} ( x^i + y^i ) d (x^j - y^j) )
$$

Choosing the [[real polarization]] spanned by $\partial_{x^i} - \partial_{y^i}$ a polarized section is function $F = F(x,y)$ such that

$$
  \iota_{\partial_{x^j} - \partial_{y^j}}(d F - \tfrac{1}{i \hbar} \tfrac{1}{4} \Theta F) = 0
$$

hence

$$
  \label{PolarizedSectionOnMorphismsOfSymplecticGroupoid}
  F(x,y)
   =
  f\left( \tfrac{x + y}{2} \right) e^{ \tfrac{1}{i \hbar} \omega\left(  \tfrac{x - y}{2} , \tfrac{x + y}{2} \right)}
  \,.
$$

=--

+-- {: .num_prop #PolarizedSymplecticGroupoidConvolutionProductOfSymplecticVectorSpaceIsMoyalStarProduct}
###### Proposition
**(polarized [[symplectic groupoid|symplectic]] [[groupoid convolution product]] of [[symplectic vector space]] is given by [[Moyal star product]])**

Given a [[symplectic vector space]] $(\mathbb{R}^{2n}, \omega)$, then
the [[groupoid convolution product]] on polarized sections (eq:PolarizedSectionOnMorphismsOfSymplecticGroupoid) on its [[symplectic groupoid]] (def. \ref{SymplecticGroupoidOfSymplecticVectorSpace}), given by [[convolution product]] followed by averaging ([[integration]]) over the polarization [[fiber]], is given by the [[star product]] (def. \ref{StarPoduct}) for the corresponding [[Poisson tensor]] $\pi \coloneqq \omega^{-1}$, in that

$$
  \begin{aligned}
    \int \int
      F(x,t) G(t,y) \, d^{2n} t \, d^{2n} (x-y)
    & =
    (f \star_\pi g)((x+y)/2)
  \end{aligned}
  \,.
$$

=--

([Weinstein 91, p. 446](Moyal+deformation+quantization#Weinstein91), [Garcia-Bondia & Varilly 94, section V](Moyal+deformation+quantization#GBV))

+-- {: .proof}
###### Proof

We compute as follows:

$$
  \begin{aligned}
    \int \int
      F(x,t) G(t,y) \, d^{2n} t \, d^{2n} (x-y)
    & \coloneqq
    \int \int f((x + t)/2) g( (t + y)/2 )
      e^{ \tfrac{1}{i \hbar} \tfrac{1}{4} \omega( x-t, x+t ) + \tfrac{1}{i \hbar} \tfrac{1}{4} \omega(t-y, t + y)  }
    \, d^{2n} t
    \, d^{2n} (x-y)
    \\
    & =
    \int \int
      f(t/2) g( (t - (x - y))/2 )
      e^{ \tfrac{1}{i \hbar} \tfrac{1}{4} \omega( (x+y) + (x - y) - t, t ) + \tfrac{1}{i \hbar} \tfrac{1}{4} \omega(t-(x+y), t - (x-y))  }
    \, d^{2n} t
    \, d^{2n} (x-y)
    \\
    & =
    \int \int
      f(t/2) g( \tilde t / 2)
      e^{ \tfrac{1}{i \hbar} \tfrac{1}{4} \omega( (x+y) - \tilde t, t ) - \tfrac{1}{i \hbar} \tfrac{1}{4} \omega((x+y)-t, \tilde t)  }
    \, d^{2n} t
    \, d^{2n} \tilde t
    \\
    & =
    \int \int
      f(t) g( \tilde t )
      e^{ \tfrac{1}{i \hbar} \tfrac{1}{4} \omega( (x+y) - 2 \tilde t, 2 t ) - \tfrac{1}{ii \hbar} \tfrac{1}{4} \omega((x+y)- 2 t, 2 \tilde t)  }
    \, d^{2n} t
    \, d^{2n} \tilde t
    \\
    \\
    & =
    \int \int
       f(t) g(\tilde t )
      e^{ \tfrac{1}{i \hbar} \omega\left( \tfrac{1}{2}(x+y) - \tilde t, \tfrac{1}{2}(x + y) - t  \right)}
     \, d^{2n} t
     \, d^{2n} \tilde t
    \\
    & =
    (f \star_\omega g)((x+y)/2)
  \end{aligned}
$$

The first line just unwinds the definition of polarized sections from def. \ref{SymplecticGroupoidOfSymplecticVectorSpace}, the following lines each implement a [[change of integration variables]] and
fnally in the last line we used prop. \ref{IntegralRepresentationOfStarProduct}.

=--

## Examples

### General examples

Some examples of star products as in def. \ref{StarPoduct}:

+-- {: .num_example}
###### Example

If $\pi = 0$ in def. \ref{StarPoduct}, then the star product $\star_0 = \cdot$ is the plain pointwise product.

=--

+-- {: .num_example #MoyalStarProduct}
###### Example
**([[Moyal star product]])**

If the tensor $\pi$ in def. \ref{StarPoduct} is skew-symmetric, it may be regarded as a constant [[Poisson tensor]] 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]] $(V,\pi)$.

=--

### Wick algebras of normal ordered products

#### Finite dimensional

+-- {: .num_defn #AlmostKaehlerVectorSpace}
###### Definition
**([[Kähler vector space]])**

An _[[Kähler vector space]]_ is a [[real vector space]] $V$ equipped with a [[linear complex structure]] $J$ as well as two [[bilinear forms]] $\omega, g \;\colon\; V \otimes_{\mathbb{R}} V \longrightarrow \mathbb{R}$ such that the following equivalent conditions hold:

1. $\omega(J v, J w) = \omega(v,w)$ and $g(v,w) = \omega(v,J w)$;

1. with $V$ regarded as a [[smooth manifold]] and with $\omega, g$ regarded as constant [[tensors]], then $(V, \omega, g)$ is an [[almost Kähler manifold]].

=--

+-- {: .num_example #StandardAlmostKaehlerVectorSpaces}
###### Example
**(standard [[Kähler vector spaces]])**

Let $V \coloneqq \mathbb{R}^2$ equipped with the [[complex structure]] $J$ which is given by the canonical identification $\mathbb{R}^2 \simeq \mathbb{C}$, hence, in terms of the canonical [[linear basis]] $(e_i)$ of $\mathbb{R}^2$, this is

$$
  J = (J^i{}_j)
  \coloneqq
  \left(
    \array{
      0 & -1
      \\
      1 & 0
    }
  \right)
  \,.
$$

Moreover let

$$
  \omega = (\omega_{i j}) \coloneqq \left( \array{0 & 1 \\ -1 & 0} \right)
$$

and

$$
  g = (g_{i j}) \coloneqq \left( \array{ 1 & 0 \\ 0 & 1}  \right)
  \,.
$$

Then $(V, J, \omega, g)$ is a [[Kähler vector space]] (def. \ref{AlmostKaehlerVectorSpace}).

The corresponding [[Kähler manifold]] is $\mathbb{R}^2$ regarded as a [[smooth manifold]] in the standard way and equipped with the [[bilinear forms]] $J, \omega g$ extended as constant rank-2 [[tensors]] over this manifold.

If we write

$$
  x,y \;\colon\; \mathbb{R}^2 \longrightarrow \mathbb{R}
$$

for the standard [[coordinate functions]] on $\mathbb{R}^2$ with

$$
  z \coloneqq x + i y \;\coloneqq\; \mathbb{R}^2 \to \mathbb{C}
$$

and

$$
 \overline{z} \coloneqq x - i y \;\coloneqq\; \mathbb{R}^2 \to \mathbb{C}
$$

for the corresponding complex coordinates, then this translates to

$$
  \omega
  \in
  \Omega^2(\mathbb{R}^2)
$$

being the [[differential 2-form]] given by

$$
  \begin{aligned}
    \omega
    & =
    d x \wedge d y
    \\
    & =
    \tfrac{1}{2i} d z \wedge d \overline{z}
  \end{aligned}
$$

and with [[Riemannian metric]] [[tensor]] given by

$$
  g = d x \otimes d x + d y \otimes d y
  \,.
$$

The [[Hermitian form]] is given by

$$
  \begin{aligned}
    h
    & =
    g - i \omega
    \\
    & =
    d z \otimes d \overline{z}
  \end{aligned}
$$

=--

(for more see at _[[Kähler vector space]]_ [this example](Kähler+vector+space#StandardAlmostKaehlerVectorSpaces)).

+-- {: .num_defn #WickAlgebraOfAlmostKaehlerVectorSpace}
###### Definition
**([[Wick algebra]] of a [[Kähler vector space]])**

Let $(\mathbb{R}^{2n},\sigma, g)$ be a [[Kähler vector space]] (def. \ref{AlmostKaehlerVectorSpace}). Then its _Wick algebra_ is the [[formal power series]] vector space $\mathbb{C}[ [ \mathbb{R}^{2n}  ] ] [ [ \hbar ] ]$ equipped with the [[star product]] (def. \ref{StarPoduct})
which is given by the [[bilinear form]]

$$
  \label{InStarProductTensorInvertingHermitianForm}
  \pi \coloneqq \tfrac{i}{2} \omega^{-1} + \tfrac{1}{2} g^{-1}
  \,,
$$

hence:

$$
  \begin{aligned}
    A_1 \star_\pi A_2
    & \coloneqq
    ((-)\cdot (-)) \circ \exp \left( \hbar\underoverset{k_1, k_2 = 1}{2 n}{\sum}\pi^{a b} \partial_a \otimes  \partial_b \right)  (A_1 \otimes A_2)
    \\
    & =
    A_1 \cdot A_2 + \hbar \underoverset{k_1, k_2 = 1}{2n}{\sum}\pi^{k_1 k_2}(\partial_{k_1} A_1) \cdot (\partial_{k_2} A_2)
    + \cdots
  \end{aligned}
$$

=--

(e.g. [Collini 16, def. 1](#Collini16))

+-- {: .num_prop #StarProductAlgebraOfKaehlerVectorSpaceIsStarAlgebra}
###### Proposition
**([[star product]] [[associative algebra|algebra]] of [[Kähler vector space]] is [[star-algebra]])**

Under [[complex conjugation]] the [[star product]] $\star_\pi$ of a [[Kähler vector space]] structure (def. \ref{WickAlgebraOfAlmostKaehlerVectorSpace}) is a [[star algebra]] in that for all $A_1, A_2 \in \mathbb{C}[ [\mathbb{R}^{2n}] ][ [\hbar] ]$ we have

$$
  \left( A_1 \star_\pi A_2 \right)^\ast
  \;=\;
  A_2^\ast \star_\pi A_1^\ast
$$

=--

+-- {: .proof}
###### Proof

This follows directly from that fact that in $\pi = \tfrac{i}{2} \omega^{-1} + \tfrac{1}{2} g^{-1}$ the [[imaginary part]] coincides with the skew-symmetric part, so that

$$
  \begin{aligned}
    (\pi^\ast)^{a b}
    & =
    -\tfrac{i}{2} (\omega^{-1})^{a b}
    + \tfrac{1}{2} (g^{-1})^{a b}
    \\
    & =
    \tfrac{i}{2} (\omega^{-1})^{b a}
    +
    \tfrac{1}{2} (g^{-1})^{b a}
    \\
    & = \pi^{b a}
    \,.
  \end{aligned}
$$

=--

+-- {: .num_example #WickAlgebraOfASingleMode}
###### Example
**([[Wick algebra]] of a single mode)**

Let $V \coloneqq \mathbb{R}^2 \simeq Span(\{x,y\})$ be the standard [[Kähler vector space]] according to example \ref{StandardAlmostKaehlerVectorSpaces}, with canonical coordinates denoted $x$ and $y$. We discuss its Wick algebra according to def. \ref{WickAlgebraOfAlmostKaehlerVectorSpace} and show that this reproduces the traditional definition of products of "normal ordered" operators as [above](#Idea).

To that end, consider the complex linear combination of the coordinates to the canonical complex coordinates

$$
  z \;\coloneqq\; x + i y
  \phantom{AAA}
    \text{and}
  \phantom{AAA}
  \overline{z} \coloneqq x - i y
$$

which we use in the form

$$
  a^\ast \;\coloneqq\; \tfrac{1}{\sqrt{2}}(x + i y)
  \phantom{AAA}
  \text{and}
  \phantom{AAA}
  a \;\coloneqq\; \tfrac{1}{\sqrt{2}}(x - i y)
$$

(with "$a$" the traditional symbol for the _[[amplitude]]_ of a field mode).

Now

$$
  \omega^{-1}
  =
  \frac{\partial}{\partial y} \otimes \frac{\partial}{\partial x}
  -
  \frac{\partial}{\partial x} \otimes \frac{\partial}{\partial y}
$$

$$
  g^{-1}
  =
  \frac{\partial}{\partial x}
  \otimes
  \frac{\partial}{\partial x}
  +
  \frac{\partial}{\partial y}
  \otimes
  \frac{\partial}{\partial y}
$$

so that with

$$
  \frac{\partial}{\partial z}
  =
  \tfrac{1}{2}
  \left(
     \frac{\partial}{\partial x}
     -i
     \frac{\partial}{\partial y}
  \right)
  \phantom{AAAA}
  \frac{\partial}{\partial \overline{z}}
  =
  \frac{1}{2}
  \left(
    \frac{\partial}{\partial x}
    +
    i
    \frac{\partial}{\partial y}
  \right)
$$

we get

$$
  \begin{aligned}
    \tfrac{i \hbar}{2}\omega^{-1} + \tfrac{\hbar}{2} g^{-1}
    & =
    2 \hbar
    \frac{\partial}{\partial \overline{z}}
      \otimes
    \frac{\partial}{\partial z}
    \\
    & =
    \hbar \frac{\partial}{\partial a} \otimes \frac{\partial }{\partial a^\ast}
  \end{aligned}
$$

Using this, we find the [[star product]]

$$
  A \star_\pi B
  \;=\;
  prod \circ \exp\left( \hbar \frac{\partial}{\partial a} \otimes \frac{\partial }{\partial a^\ast}  \right)
$$

to be as follows (where we write $(-)\cdot (-)$ for the plain commutative product in the [[formal power series algebra]]):

$$
  \begin{aligned}
    a \star_\pi a & = a \cdot a
    \\
    a^\ast \star_\pi a^\ast & = a^\ast \cdot a^\ast
    \\
    a^\ast \star_\pi a & = a^\ast \cdot a
    \\
    a \star_\pi a^\ast
    & =
    a \cdot a^\ast + \hbar
  \end{aligned}
$$

and so forth, for instance

$$
  \array{
    (a \cdot a ) \star_\pi (a^\ast \cdot a^\ast)
    & =
    a^\ast \cdot a^\ast \cdot a \cdot a
    +
    4 \hbar a^\ast \cdot a
    +
    \hbar^2
  }
$$

If we instead indicate the commutative pointwise product by colons and the star product by plain juxtaposition

$$
  :f g: \;\coloneqq\; f \cdot g
  \phantom{AAAA}
  f g \;\coloneqq\; f \star_\pi
$$

then this reads

$$
  \array{
    :a a: \, :a^\ast a^\ast:
    & =
    : a^\ast a^\ast a  a :
    +
    4 \hbar \, : a^\ast a :
    +
    \hbar^2
  }
$$

This is the way the _[[Wick algebra]]_ with its _[[operator product]]_ $\star_\pi$  and _[[normal-ordered product]]_ $:-:$ is traditionally presented.

=--

#### Infinite-dimensional

+-- {: .num_example #WickAlgebras}
###### Example
**([[normal-ordered products]] and [[time-ordered products]])**

In cases where $V$ is _not_ [[finite number|finite]] [[dimension|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. \ref{StarPoduct} still may be interpreted as [[products of distributions]], but these only exist if the [[wavefront sets]] satisfy [[Hörmander's criterion]].

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 (violation of [[Hörmander's criterion]]), 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. \ref{SymmetricContribution} 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 $\star_\pi$ does not exist, and $\pi = \tfrac{1}{2}\pi + \alpha$ is a [[Wightman propagator]], so that $\star_{\pi}$ does exist on [[microcausal functionals]]: it is the "[[normal-ordered product]]" of quantum observables.

=--

[[!include Wick algebra -- table]]

## Related concepts

* [[deformation theory]]

## References

* {#Collini16} [[Giovanni Collini]], _Fedosov Quantization and Perturbative Quantum Field Theory_ ([arXiv:1603.09626](https://arxiv.org/abs/1603.09626))

In [[perturbative quantum field theory]]:

* [[Michael Dütsch]], section 2.1 of _[[From classical field theory to perturbative quantum field theory]]_

[[!redirects star products]]

[[!redirects star-product]]


star-algebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A __$*$-algebra__ is an [[associative algebra|algebra]] $A$ ([[associative algebra|associative]] or [[nonassociative algebra|non-associative]]) equipped with an [[anti-involution]], meaning a map $(-)^\ast:A \to A$ such that

* for all elements $a \in A$ and $b \in A$, $(a b)^\ast = b^\ast a^\ast$

* for all elements $a \in A$, $(a^\ast)^\ast = a$

* $1^\ast = 1$

## Definition

In more detail, begin with a [[commutative ring]] (often a [[field]], but possibly just a [[rig]]) $K$ equipped with an [[involution]] (a [[homomorphism]] whose square is the [[identity morphism|identity]]), written $x \mapsto \bar{x}$.  (The usual example for $K$ is the field of [[complex numbers]] with involution given by [[complex conjugation]], but the concept of $*$-algebra makes sense in more general contexts.  (An example is given by *any* commutative ring $K$ with trivial involution $\bar{x} \coloneqq x$.)

Then a __$K$-$*$-algebra__ (a $*$-algebra over $K$) is a $K$-[[module]] $A$ equipped with a $K$-[[bilinear map]] $A\times A \to A$, written as multiplication (and often assumed to be [[associativity|associative]]) and a $K$-[[antilinear map]] $A \to A$, written as $x \mapsto x^*$, such that

* $ (x^\ast)^\ast = x$ for all $x$ in $A$ (so we have an [[involution]] on the underlying $K$-module), and

* $(x y)^* = y^* x^*$ for all $x,y$ in $A$ (so it is an [[anti-involution]] on $A$ itself)

* $1^\ast = 1$.

The hypothesis that the anti-involution is $K$-antilinear means that $(r x)^* = \overline{r} x^*$ for all $r$ in $K$ and all $x$ in $A$ (as well as $(x + y)^* = x^* + y^*$).

If a $K$-$*$-algebra $A$ is itself commutative, then it is in particular a commutative ring with involution, and one can consider $A$-$*$-algebras as well.  On the other hand, a commutative ring with involution is simply a commutative $*$-algebra over the ring of [[integers]] (with trivial involution), and similarly for rigs and [[natural numbers]].

### $*$-Rings

A __$*$-ring__ is simply a $*$-algebra over the ring of [[integers]] (with trivial involution).  Similarly, a __$*$-rig__ is a $*$-algebra over the rig of [[natural numbers]].

Arguably, when we began this article with a commutative ring $K$ equipped with involution, we should have begun it with a ring with *anti*-involution instead.  However, since the ring (or rig) is commutative, there is no difference.

### Banach $*$-algebras

When $K$ is the field $\mathbb{C}$ of [[complex numbers]] (or the field $\mathbb{R}$ of [[real numbers]], with trivial involution), we can additionally ask that the $*$-algebra be a [[Banach algebra]]; then it is a __Banach $*$-algebra__.  Special cases of this are

* $C^*$-[[C-star-algebra|algebras]] (aka $B^*$-algebras)

* and [[von Neumann algebras]] (aka $W^*$-algebras)

Arguably, one should require that the map $*$ be an [[isometry]] (which follows already if it is required to be [[short map|short]]); some authors require this and some don\'t.  However, this is automatic in the case of $C^*$-algebras (and hence also von Neumann algebras).

## Examples

### General
 {#General}

\begin{example}\label{TrivialStarStructure}
**(trivial star-structure)**\linebreak
Any plain [[algebra]] becomes a star-algebra by equipping it with the [[identity morphism|identity]] anti-involution.
\end{example}

\begin{example}\label{ComplexConjugation}
**([[complex conjugation]])**\linebreak
  The [[complex numbers]] -- regarded as an [[associative algebra]] over the [[real numbers]] -- form a star-algebra with anti-involution (which here  is an involution, since the product is [[commutative algebra|commutative]]) given by [[complex conjugation]].
\end{example}

\begin{example}
**([[Cayley-Dickson construction]])**
The [[Cayley-Dickson construction]] takes any star-algebra over the [[real numbers]] to a new star algebra.

Applied to the [[real numbers]] trivially regarded as a star-algebra (via Example \ref{TrivialStarStructure}), this yields, successively, the star-algebras of

* [[real numbers]] (Example \ref{TrivialStarStructure})

* [[complex numbers]] (Example \ref{ComplexConjugation})

* [[quaternions]]

* [[octonions]]

(which are the four [[real normed division algebras]]) and then further the

* [[sedenions]]

* ...

In each case the star-operation may be thought of as [[complex conjugation]], given by changing the sign of the [[imaginary part]] and keeping the [[real part]] intact.

\end{example}

\begin{example}
**(operators on a [[Hilbert space]])**
The algebra of [[continuous operators]] on a [[Hilbert space]] is a star-algebra, with the [[Hermitian adjoint]] as the anti-involution.
\end{example}

\begin{example}
**([[C-star algebras]])**
A _[[C-star algebra]]_ is a star-algebra where the anti-involution is compatible with the [[norm]] of the underlying [[Banach algebra]].
\end{example}

### Involutive Hopf algebras
 {#InvolutiveHopfAlgebras}

\begin{example}\label{InvolutiveHopfAlgebrasAreStarAlgebras}
**([[involutive Hopf algebras]] are [[star-algebras]])** \linebreak
  Any [[involutive Hopf algebra]] is a star-algebra, with star-involution given by the [[antipode]] (by [this Prop.](Hopf+algebra#AntipodeIsAnAntihomomorphism)).
\end{example}

\begin{example}
**(groupoid algebras are star-algebras)**
  A [[group algebra]] and, more generally, a [[groupoid convolution algebra]], is a star-algebra, with the star-[[involution]] given by pullback along the [[inverse|inversion]] operation of the groupoid.

Yet more generally, the [[category convolution algebra]] of a [[dagger-category]] is a $*$-algebra, with the involution being the pullback along the $\dagger$ operation.

All these are [[involutive Hopf algebras]] (since taking [[inverses]] and taking [[dagger-category|dagger-operations]] squares to the identity) and as such are special cases of Example \ref{InvolutiveHopfAlgebrasAreStarAlgebras}
\end{example}

\begin{example}\label{StarAlgebraOfHorizontalChordDiagrams}
**(star-algebra of [[horizontal chord diagrams]])** \linebreak
The algebra of [[horizontal chord diagrams]] is a star-algebra under reversal of orientation of strands (see [here](horizontal+chord+diagram#StarAlgebraStructure), [CSS 21, Prop. 2.9](#CSS21)).

Since [[horizontal chord diagrams are the homology of the loop space of configuration space]] and the [[homology of a loop space]] is an [[involutive Hopf algebra]], this is a special case of Example \ref{InvolutiveHopfAlgebrasAreStarAlgebras}.
\end{example}

## Generalizations

This concept could be generalized from the category of $R$-modules to any [[monoidal category]]:

Let $(C, I, \otimes)$ be a [[monoidal category]] and let $(A, e:I \to A, \pi:A \otimes A \to A)$ be a [[unital algebra object]] in $C$. Then $A$ is a **$*$-algebra object** [[internalization|in]] $C$ if it comes with a morphism $\iota:A \to A$ such that

* $\iota \circ \iota = \mathrm{id}_A$

* $\iota \circ e = e$

* for all morphisms $a:I \to A$ and $b:I \to A$,

$$\iota \circ \pi \circ (a \otimes b) = \pi \circ ((\iota \circ b) \otimes (\iota \circ a))$$

In the [[category of sets]], $*$-algebra objects are called **[[anti-involution|anti-involutive]] [[unital magmas]]**, and in the [[category of abelian groups]], associative $*$-algebra objects are called **[[*-rings]]**.

## Related concepts

* [[real part]], [[imaginary part]]

* [[anti-linear map]], [[anti-dual linear space]]

* [[Hermitian form]]

* [[state on a star-algebra]]

* [[star-representation]]

[[!include states and observables -- content]]

## References

See also:

* Wikipedia, _[star-algebra](https://en.wikipedia.org/wiki/*-algebra)_

Discussion for [[group algebras]]:

* Antonio Giambruno, Cesar Polcino Milies and Sudarshan K. Sehgal, *Star-group identities on units of group algebras: The non-torsion case*, Forum Mathematicum Volume 30 Issue 1 ([doi:10.1515/forum-2016-0266](https://doi.org/10.1515/forum-2016-0266))

Discussion of the Example \ref{StarAlgebraOfHorizontalChordDiagrams} of [[horizontal chord diagrams]]:

* {#CSS21} [[David Corfield]], [[Hisham Sati]], [[Urs Schreiber]]: *[[schreiber:Fundamental weight systems are quantum states]]* ([arXiv:2105.02871](https://arxiv.org/abs/2105.02871))

[[!redirects star-algebra]]
[[!redirects star-algebras]]
[[!redirects star algebra]]
[[!redirects star algebras]]
[[!redirects *-algebra]]
[[!redirects *-algebras]]
[[!redirects * algebra]]
[[!redirects * algebras]]

[[!redirects star-ring]]
[[!redirects star-rings]]
[[!redirects star ring]]
[[!redirects star rings]]
[[!redirects *-ring]]
[[!redirects *-rings]]
[[!redirects * ring]]
[[!redirects * rings]]

[[!redirects star-rig]]
[[!redirects star-rigs]]
[[!redirects star rig]]
[[!redirects star rigs]]
[[!redirects *-rig]]
[[!redirects *-rigs]]
[[!redirects * rig]]
[[!redirects * rigs]]

[[!redirects Banach star-algebra]]
[[!redirects Banach star-algebras]]
[[!redirects Banach star algebra]]
[[!redirects Banach star algebras]]
[[!redirects Banach *-algebra]]
[[!redirects Banach *-algebras]]
[[!redirects Banach * algebra]]
[[!redirects Banach * algebras]]

[[!redirects star-algebra object]]
[[!redirects star-algebra objects]]
[[!redirects star algebra object]]
[[!redirects star algebra objects]]
[[!redirects *-algebra object]]
[[!redirects *-algebra objects]]
[[!redirects * algebra object]]
[[!redirects * algebra objects]]


star-autonomous category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--

#Contents
* table of contents
{:toc}

## Idea

There are many ways to describe a $*$-autonomous category; here are a few:

* it is a [[monoidal category]] in which all objects have "duals", but in a weaker sense than in a [[compact closed category]].
* it is a [[closed monoidal category]] in which the [[internal-hom]] can be expressed in terms of the tensor product in a particular way.
* it is a [[linearly distributive category]] in which all objects have "duals" in an appropriate linearly distributive sense.
* it is a semantics for classical multiplicative [[linear logic]].
* it is a representable [[star-polycategory]].
* it is a sort of categorified [[Frobenius algebra]].

In particular, it has two monoidal structures $\otimes$ and $\parr$, as in a linearly distributive category.  However, because of the dualization operation $(-)^\ast$, each of them determines the other by a "multiplicative [[de Morgan's laws|de Morgan]] duality": $A\parr B \cong (A^\ast \otimes B^\ast)^\ast$.  Thus, in the definition we only have to refer to one monoidal structure.

A $\ast$-autonomous category is a special case of the notion of [[star-autonomous pseudomonoid]] (a.k.a. Frobenius pseudomonoid, since it categorifies a [[Frobenius algebra]]) in a [[monoidal bicategory]].

(Regarding the use of "autonomous", this was once used as a bare adjective to describe a [[closed monoidal category]], or sometimes a [[compact closed monoidal category]], but is rarely used in this way today. It has been suggested that in these cases with internal-hom objects the category is autonomous or "sufficient unto itself" without needing hom-sets, or suchlike.)

## Definition ##

There are two useful equivalent formulation of the definition

+-- {: .num_defn #DefByDualizingObject}
###### Definition

A _$*$-autonomous category_  is a [[symmetric monoidal
category|symmetric]] [[closed monoidal category]] $\langle
C,\otimes, I,\multimap\rangle$ with a _[[dualizing object in a closed category|global dualizing object]]_: an object
$\bot$ such that the canonical morphism
$$ d_A \;\colon\; A \to (A \multimap \bot) \multimap \bot ,$$
which is the transpose of the [[evaluation map]]
$$ ev_{A,\bot} \;\colon\; (A \multimap \bot) \otimes A \to \bot\,,$$
is an [[isomorphism]] for all $A$.  (Here, $\multimap$ denotes the [[internal hom]].)

=--

+-- {: .num_defn #DefByDualization}
###### Definition

A _$*$-autonomous category_  is a [[symmetric monoidal category]] $\mathcal{C}$
equipped with a [[full and faithful functor|full and faithful]] [[functor]]

$$
  (-)^\ast \colon \mathcal{C}^{op} \longrightarrow \mathcal{C}
$$

such that there is a [[natural isomorphism]]

$$
  \mathcal{C}(A \otimes B, C^\ast) \simeq \mathcal{C}(A, (B\otimes C)^\ast)
  \,.
$$

=--

+-- {: .num_prop}
###### Proposition

These two definitions, def. \ref{DefByDualizingObject}
and def. \ref{DefByDualization}, are indeed equivalent.

=--

+-- {: .proof}
###### Proof

Given def. \ref{DefByDualizingObject},
define the dualization functor as the [[internal hom]] into the [[dualizing object in a closed category|dualizing object]]

$$
  (-)^ \ast  \coloneqq (-)\multimap\bot
  \,.
$$

Then the morphism $d_A$
is natural in $A$, so that there is a [[natural isomorphism]]
$d:1\Rightarrow(-)^{**}$.  We also have

$$\begin{aligned}
\hom(A\otimes B,C^*)& = \hom(A\otimes B, C\multimap\bot) \\
& \cong \hom((A\otimes B)\otimes C,\bot) \\
& \cong \hom(A\otimes(B\otimes C), \bot) \\
& \cong \hom(A,(B\otimes C)\multimap\bot) \\
& = \hom(A,(B\otimes C)^*)
\end{aligned}$$

This yields the structure of def. \ref{DefByDualization}.

Conversely, given the latter then the [[dualizing object]] $\bot$ is defined as the dual of the [[tensor unit]] $\bot \coloneqq I^*$ and the [[internal hom]] by $A \multimap B \coloneqq A^* \parr B$ (where $\parr$ is defined as below).

=--

(See [this discussion](https://github.com/punkdit/categories/blob/master/www.mta.ca/cat-dist/archive/2007/07-12#L2488-L2574) on the [categories mailing list](https://www.mta.ca/~cat-dist/) for alternative definitions and remarks about coherence.)

+-- {: .num_remark}
###### Remark

A $\ast$-autonomous category in which the [[tensor product]] is compatible with duality in that there is a [[natural isomorphism]]

$$
  (A \otimes B)^\ast \simeq A^\ast \otimes B^\ast
$$

is a [[compact closed category]].  This follows from a stronger result of Dold and Puppe [DP83]{#DP83}.
=--

More generally, even if the $\ast$-autonomous category is not compact closed, then by this linear "[[de Morgan duality]]" the tensor product induces a second binary operation

$$
  A \parr B \coloneqq  (A^\ast \otimes B^\ast)^\ast
  \,.
$$

making it into a [[linearly distributive category]].  Here the notation on the left is that used in [[linear logic]], see below at _[Properties -- Internal logic](#InternalLogic)_.

A general $\ast$-autonomous category can be thought of as like a [[compact closed category]] in which the [[unit]] and [[counit]] of the [[dual objects]] refer to two different tensor products: we have $\top \to A \parr A^*$ but $A^* \otimes A \to \bot$, where $(\otimes,\top)$ and $(\parr,\bot)$ are two different monoidal structures.  The necessary relationship between two such monoidal structures such that this makes sense, i.e. such that the [[triangle identities]] can be stated, is encoded by a linearly distributive category; then an $\ast$-autonomous category is precisely a linearly distributive category in which all such "mixed duals" (or "negations") exist.

=--

## Functors and transformations

It may not be clear from the above definitions what the appropriate notion of "$\ast$-autonomous functor" is.  In fact, from at least one perspective it suffices to consider ordinary [[lax monoidal functors]], at least in the symmetric case; no interaction with the $\ast$-autonomy is required.

Let $\ast Aut$ denote the full sub-2-category of $SymMonCat_{lax}$ on the (symmetric) $\ast$-autonomous categories; hence its morphisms and 2-cells are lax symmetric monoidal functors and monoidal natural transformations.  Let $SymLinDist$ denote the 2-category of symmetric [[linearly distributive categories]], symmetric linear functors, and linear transformations (defined at [[linearly distributive category]]).

+-- {: .un_theorem}
###### Theorem
There is a functor $\ast Aut \to SymLinDist$ that is 2-fully-faithful, i.e. an equivalence on hom-categories, and has both a left and a right [[2-adjoint]].
=--
+-- {: .proof}
###### Proof
A linearly distributive functor consists of a functor $F_\otimes$ that is lax monoidal for $\otimes$ and a functor $F_\parr$ that is lax monoidal for $\parr$.  Recalling that $\parr$ is defined in a $\ast$-autonomous category as $A\parr B = (A^* \otimes B^*)^*$, if $F$ is a lax monoidal functor between $\ast$-autonomous categories we define $F_\otimes = F$ and $F_\parr(A) = (F(A^*))^*$.  See [Cockett-Seely 1999](#CockettSeely99) for details.
=--

In the non-symmetric case, we need to additionally require of an "$\ast$-autonomous functor" that $^*(F(A^*)) \cong (F({}^*A))^*$, and define $F_\parr$ to be their common value.

On the other hand, if we consider instead "Frobenius linear functors" between linearly distributive categories, then such functors necessarily preserve duals.  Thus, the corresponding notion of $\ast$-autonomous functor would need to preserve the dualization functors as well, $F(A^*) \cong (F(A))^*$.

## Properties

### Internal logic
 {#InternalLogic}

The [[internal logic]] of star-autonomous categories is the multiplicative fragment of [[classical linear logic]], conversely star-autonomous categories are the [[categorical semantics]] of [[classical linear logic]] ([Seely 89, prop. 1.5](#Seely89)). See also at _[[relation between type theory and category theory]]_.

## Examples ##

* A simple example of a $*$-autonomous category is the category of [[finite-dimensional vector space]]s over some field $k$.  In this case $k$ itself plays the role of the dualizing object, so that for an f.d. vector space $V$, $V^*$ is the usual [[dual vector space|dual space]] of linear maps into $k$.

* More generally, any [[compact closed category]] is $*$-autonomous with the unit $I$ as the dualizing object.

* A more interesting example of a $*$-autonomous category is the category of [[sup-lattice]]s and sup-preserving maps (= left adjoints). Clearly the poset of sup-preserving maps $hom(A, B)$ is itself a sup-lattice, so this category is closed. The free sup-lattice on a poset $X$ is the internal hom of posets $[X^{op}, \Omega]$; in particular the poset of truth values $\Omega$ is a unit for the closed structure. Define a duality $ (-)^* $ on sup-lattices, where $ X^* = X^{op} $ is the opposite poset (inf-lattices are sup-lattices), and where $ f^*: Y^* \to X^* $ is the left adjoint of $f^{op}: X^{op} \to Y^{op}$. In particular, take as dualizing object $D = \Omega^{op}$. Some simple calculations show that under the tensor product defined by the formula $ (X \multimap Y^*)^* $, the category of sup-lattices becomes a $*$-autonomous category.

* Another interesting example is due to Yuri Manin: the category of [[quadratic algebra]]s. A **quadratic algebra** over a [[field]] $k$ is a graded algebra $A = T(V)/I$, where $V$ is a finite-dimensional vector space in degree 1, $T(V)$ is the tensor algebra (the free $k$-algebra generated by $V$), and $I$ is a graded ideal generated by a subspace $R \subseteq V \otimes V$ in degree 2; so $R = I_2$, and $A$ determines the pair $(V, R)$. A morphism of quadratic algebras is a morphism of graded algebras; alternatively, a morphism $(V, R) \to (W, S)$ is a linear map $f: V \to W$ such that $(f \otimes f)(R) \subseteq S$. Define the dual $A^*$ of a quadratic algebra given by a pair $(V, R)$ to be that given by $(V^*, R^\perp)$ where $R^\perp \subseteq V^* \otimes V^*$ is the kernel of the transpose of the inclusion $i: R \subseteq V \otimes V$, i.e., there is an exact sequence
$$0 \to R^\perp \to V^* \otimes V^* \overset{i^*}{\to} R^* \to 0$$
Define a tensor product by the formula
$$(V, R) \otimes (W, S) = (V \otimes W, (1_V \otimes \sigma \otimes 1_W)(R \otimes S))$$
where $\sigma: V \otimes W \to W \otimes V$ is the symmetry. The unit is the tensor algebra on a 1-dimensional space. The hom that is adjoint to the tensor product is given by the formula $A \multimap B = (A \otimes B^*)^*$, and the result is a $*$-autonomous category.

* In a similar vein, I am told that there is a $*$-autonomous category of [[quadratic operad]]s.

* Girard's [[coherence spaces]], developed as models of [[linear logic]], give an historically important example of a $*$-autonomous category.  These are closely related to a general construction of $*$-autonomous categories (and related types of categories) called [[poset-valued sets]].

* The category of [[finiteness spaces]] and their relations is $*$-autonomous.  Probably so is any category of [[arity spaces]], which includes coherence spaces and finiteness spaces.

* Hyland and Ong have given a completeness theorem for multiplicative linear logic in terms of a $*$-autonomous category of _fair games_, part of a series of work on game semantics for closed category theory (compare Joyal's interpretation of Conway games as forming a compact closed category).

* The [[Chu construction]] can be used to form many more examples of $*$-autonomous categories.

* Various [[subcategories]] of Chu constructions are also $*$-autonomous.  For instance, if [[Vect]] is the category of [[vector spaces]] over a [[field]] $k$, then $Chu(Vect,k)$ is the category of vector spaces equipped with a specified "dual" having no further structure than an evaluation map $V\otimes W\to k$.  One often wants to impose nondegeneracy conditions on this "dual", which in turn can be reflected as [[topological vector space|topological]] properties of the original space $V$.
Write $(V,V')$ for an object of $Chu(Vect,k)$ and $\langle v,w \rangle$ the evaluation of $w$ on $v$. We say that $(V,V')$ is separated if for each $v \neq 0 \in V$, there exists $v' \in V'$ such that $\langle v,v' \rangle \neq 0$ and we say that it is extensional if for each $v' \neq 0 \in V'$, there exists $v \in V$ such that $\langle v,v' \rangle \neq 0$. Then, the full subcategory $chu(Vect,k)$ of separated, extensional pairs is $*$-autonomous.

* A [[quantale]] (see there) is a $\ast$-autonomous category if it has a [[dualizing object]].

* Suppose $\langle C,\otimes, I,\multimap\rangle$ is a closed symmetric monoidal category equipped with a "pre-dualizing object" $\bot$, in the sense that the contravariant self-adjunction $(-\multimap\bot) \dashv (-\multimap\bot)$ is [[idempotent adjunction|idempotent]], i.e. the double-dualization map $A \to (A \multimap \bot) \multimap \bot$ is an isomorphism whenever $A$ is of the form $B\multimap\bot$.  (Note that idempotence is automatic if $C$ is a poset.)  Then the category of [[fixed point of an adjunction|fixed points]] of this adjunction, i.e. the full subcategory of objects of the form $B\multimap\bot$, is $*$-autonomous.  For it is closed under $\multimap$, as $(A\multimap (B\multimap \bot)) \cong (A\otimes B\multimap \bot)$, and reflective with reflector $(-\multimap\bot)\multimap\bot$, and it contains $\bot$ since $\bot\cong (I\multimap \bot)$.  Hence it is closed symmetric monoidal with tensor product $((A\otimes B)\multimap\bot)\multimap\bot$, and all its double-dualization maps are isomorphisms by assumption.  A historically important example is Girard's [[phase semantics]] of [[linear logic]].  Note that this category is a full subcategory of $Chu(C,\bot)$ closed under duality --- indeed, it is the intersection of the two embeddings of $C$ and $C^{op}$ therein --- but its tensor product is *not* the restriction of the tensor product of $Chu(C,\bot)$.

* Phase semantics is also a special case of a [[ternary frame]], which is the case of the previous example when $\langle C,\otimes, I,\multimap\rangle$ has the [[Day convolution]] structure induced from a [[promonoidal category|promonoidal]] poset.  There is also another way to obtain negation in a ternary frame involving a "compatibility relation"...

* The [[unit interval]] $[0,1]$ is a [[semicartesian monoidal category|semicartesian]] $\ast$-autonomous poset with the monoidal structure $x \otimes y = \max(x+y-1,0)$ and dualizing object $0$.  The involution is $x^\ast = 1-x$ and the dual [[multiplicative disjunction]] is $x\parr y = \min(x+y,1)$.  This is known as [[Lukasiewicz logic]] and is used in [[fuzzy logic]]; it is also a special case of a [[t-norm]].

* A summary of many different ways to construct examples is in [Hyland-Schalk](#HylandSchalk01).

## Relation to other structures

* A $\ast$-autonomous category is a [[linearly distributive category]] with the cotensor product $x\parr y = (x^* \otimes y^*)^*$.  Conversely, any linearly distributive category with "duals" in the appropriate sense is $\ast$-autonomous.

* Since a linearly distributive category is a representable [[polycategory]], a $\ast$-autonomous category is a representable polycategory with duals.  If the duals are strictly involutive, then it is a [[*-polycategory]].

* As noted above, a [[compact closed]] category is a degenerate sort of $\ast$-autonomous category.

* It is shown in [HH13](#HH13) that a $\ast$-autonomous category that is [[traced monoidal category|traced]] must be compact closed.

* We can weaken the requirement that the canonical morphism
$ d_A: A \to (A \multimap \bot) \multimap \bot $ is an isomorphism to the one that it is a monomorphism: [[weak (star)-autonomous category]].

## References ##

The notion is originally due to:

* [[Michael Barr]], _$\ast$-Autonomous Categories_, Lecture Notes in Mathematics **752**, Springer (1979) [[doi:10.1007/BFb0064579](https://doi.org/10.1007/BFb0064579)]

See also:

* [[Michael Barr]], [[Heinrich Kleisli]], *Topological balls*, [[Cahiers|Cahiers Topologie Géométrie Différentielle Catégorique]], **40** 1 (1999) 3–20 [[numdam:CTGDC_1999__40_1_3_0](http://www.numdam.org/item/?id=CTGDC_1999__40_1_3_0)]

* [[Michael Barr]], _$\ast$-Autonomous categories: once more around the track_, Theory and Applications of Categories **6** 1 (1999) 5-24 [[tac:6-01](http://www.tac.mta.ca/tac/volumes/6/n1/6-01abs.html)]

* [[Michael Barr]], _Topological $\ast$-autonomous categories, revisited_, rewrite of TAC, 16 (2006), 700-708 ([arXiv:1609.04241](http://arxiv.org/abs/1609.04241))

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *On $\ast$-autonomous categories of topological modules*, Theory Appl. Categories, **24** 14 (2010) 278–293 [[tac:24-14](http://www.tac.mta.ca/tac/volumes/24/14/24-14abs.html)]

The relation to [[linear logic]] was first described in

* {#Seely89} [[R. A. G. Seely]],  _Linear logic, $\ast$-autonomous categories and cofree coalgebras_, _Contemporary Mathematics_ 92, 1989.  ([[SeelyLinearLogic.pdf:file]], [ps.gz](http://www.math.mcgill.ca/rags/nets/llsac.ps.gz))

further discussed in

* [[Michael Barr]], *$\ast$-Autonomous categories and linear logic*, Math. Structures Comp. Sci. **1** 2 (1991) 159–178 [[doi:10.1017/S0960129500001274](https://doi.org/10.1017/S0960129500001274), [pdf](https://www.math.mcgill.ca/barr/papers/scatll.pdf), [[Barr-AutomomousCatAndLinLogic.pdf:file]]]

* [[Michael Barr]], *Accessible categories and models of linear logic*, J. Pure Appl. Algebra **69** (1990) 219–232 [doi:10.1016/0022-4049(91)90020-3[pdf](https://www.math.mcgill.ca/barr/papers/accll.pdf), [[Barr-AccCatLinearLogic.pdf:file]]]

and a detailed review (also of a fair bit of plain monoidal category theory) is in

* [[Paul-André Melliès]], _Categorial Semantics of Linear Logic_, in _Interactive models of computation and program behaviour_, Panoramas et synthèses 27, 2009 ([pdf](http://www.pps.univ-paris-diderot.fr/~mellies/papers/panorama.pdf))

Examples from algebraic geometry are given here:

* Mitya Boyarchenko and Vladimir Drinfeld, A duality formalism in the spirit of Grothendieck and Verdier, arXiv:1108.6020 ([pdf](http://arxiv.org/pdf/1108.6020v2.pdf))

These authors call any closed monoidal category with a [[dualizing object in a closed category|dualizing object]] a **Grothendieck-Verdier category**, thanks to the examples coming from [[Verdier duality]].

Here it is explained how $*$-autonomous categories give Frobenius pseudomonads in the 2-category where morphisms are [[profunctors]]:

*  [[Ross Street]], *Frobenius monads and pseudomonoids*, J. Math. Physics **45** (2004) 3930-3948 [[pdf](http://www.math.mq.edu.au/~street/Frob.pdf)]

Examples using toplogical vector spaces are given here:

* [[Michael Barr]], *On $*$-autonomous categories of topological vector spaces*, Cahiers de topologie et géométrie différentielle catégoriques **41** 4 (2000) 243-254 [[numdan:CTGDC_2000__41_4_243_0] (http://www.numdam.org/item/CTGDC_2000__41_4_243_0.pdf)]

Relation to [[linearly distributive categories]]:

* {#CockettSeely99} [[Robin Cockett]], [[Robert Seely]], _Linearly distributive functors_ (1999) [doi:10.1016/S0022-4049(98)00110-8]

Combination with traces yields compact closure:

* {#HH13} Tamás Hajgató and [[Masahito Hasegawa]], *Traced $\ast$-autonomous categories are compact closed*, [TAC](http://www.tac.mta.ca/tac/volumes/28/7/28-07abs.html), 2013

A wide-ranging summary of different model constructions:

* {#HylandSchalk01 [[Martin Hyland]] and [[Andrea Schalk]], _Glueing and orthogonality for models of linear logic_, [pdf](http://www.cs.man.ac.uk/~schalk/publ/gomll.pdf)

The fact that a $\ast$-autonomous category for which there is a [[natural isomorphism]] $(A \otimes B)^\ast \simeq A^\ast \otimes B^\ast$ is a [[compact closed category]] has a purely [[string diagram|string diagrammatic]] proof, but it also follows from the implication (a)$\implies$ (b) of Theorem 1.3 in this paper:

* {#DP83} [[Albrecht Dold]], [[Dieter Puppe]], *Duality, Trace and Transfer*, Proceedings of the Steklov Institute of Mathematics **154** (1984) 85–103 [[mathnet:tm2435](http://mi.mathnet.ru/tm2435), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/doldpup2.pdf), [pdf](http://www.maths.ed.ac.uk/~aar/papers/doldpup2.pdf)]

According to Tobias Fritz (on the [Category Theory Community Server](https://categorytheory.zulipchat.com/#narrow/stream/229136-theory.3A-category-theory/topic/From.20*-autonomous.20to.20compact.20closed)):

>  They apparently didn't know about $\ast$-autonomous categories (which were introduced a few years prior to that), and their statement has slightly weaker assumptions in that they start with a natural isomorphism $\mathscr{C}(A \otimes B, I) \cong \mathscr{C}(A, B^*)$ only, but they also require the induced morphisms $A^* \otimes B^* \to (A \otimes B)^*$ to be isomorphisms.

See also:

* [[Michael Barr]], *Topological $\ast$-autonomous categories*, Theory Appl. Categories, 16 (2006) 700–708 [[pdf](https://www.math.mcgill.ca/barr/papers/mack.pdf), [[Barr-TopologicalStarAutonomous.pdf:file]]]

* [[Michael Barr]], *Topological $\ast$-autonomous categories, revisited*, Tbilisi Math. J. **10** 3 (2017) 51–64 [[arXiv:1609.04241](https://arxiv.org/abs/1609.04241)]

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *The $\ast$-autonomous category of uniform sup semi-lattices*,  Theory and Applications of Categories **27** 11 (2012) 222–241 [[tac:27-11](http://www.tac.mta.ca/tac/volumes/27/11/27-11abs.html)]

[[!redirects star-autonomous categories]]
[[!redirects *-autonomous category]]
[[!redirects *-autonomous categories]]
[[!redirects star-autonomous]]
[[!redirects *-autonomous]]


star-category

# $*$-categories
* table of contents
{: toc}

## Idea

The notion of $*$-category combines that of a $\dagger$-[[dagger-category|category]] and of an $\mathbb{C}$-[[algebroid]] (i.e. a $\mathbb{C}$-[[linear category]]).  Thus it might reasonably be called simply a *$\mathbb{C}$-linear $\dagger$-category*.  However, $*$-categories were popular in [[AQFT]] for a while, while the notion of [[dagger-category]] was introduced in the context of [[quantum information theory]], so for historical reasons we have two names.

## Definition

+-- {: .un_defn}
###### Definition
A $*$-category is a [[category]] that is a $\dagger$-[[dagger-category|category]] and a $\mathbb{C}$-[[algebroid]] such that the dagger operation is conjugate linear.
=--

In a $*$-category, the dagger operation $\dagger$ is synonymously denoted $*$.

## Examples

See [[DHR category]].

## Related concepts

* [[C-star-category]]

[[!redirects star-category]]
[[!redirects star-categories]]
[[!redirects star category]]
[[!redirects star categories]]
[[!redirects *-category]]
[[!redirects *-categories]]
[[!redirects linear dagger-category]]
[[!redirects linear dagger-categories]]
[[!redirects linear dagger category]]
[[!redirects linear dagger categories]]
[[!redirects dagger-linear category]]
[[!redirects dagger-linear categories]]
[[!redirects dagger linear category]]
[[!redirects dagger linear categories]]
[[!redirects linear †-category]]
[[!redirects linear †-categories]]
[[!redirects †-linear category]]
[[!redirects †-linear categories]]


star-monoid

A $*$-monoid is a [[monoid]] with a compatible [[involution]].  The compatibility requirement is this:

$$ (a b)^* = b^* a^* .$$

These generalize to $*$-[[star-ring|rings]], $C^*$-[[C-star-algebra|algebras]], [[dagger categories]], etc.

[[!redirects star-monoid]]
[[!redirects star-monoids]]
[[!redirects *-monoid]]
[[!redirects *-monoids]]


star-polycategory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A **star-polycategory** is to a [[star-autonomous category]] what a [[multicategory]] is to a [[monoidal category]]. It is a [[polycategory]] in which
every object has a dual. As such it is a model of classical linear logic with no connectives except negation.

## Definition

There are a number of candidate definitions, depending on what we make strict.

### Definition in terms of duals

Let $A$ be an object in a symmetric polycategory. A _dual_ for $A$ comprises an object $A^*$ together with a morphism $Hom(A,A^*;-)$ making the induced composition $Hom(\Gamma;A,\Delta)\to Hom(\Gamma,A^*;\Delta)$ a bijection for all $\Gamma$ and $\Delta$.

A symmetric _star-polycategory_ is a symmetric polycategory in which every object has a dual.

### Strict involution

In any star-polycategory there is an isomorphism $A^{**}\cong A$ for all $A$. We could also require equality, $A^{**}=A$. This holds in many concrete examples, and it holds in syntactic examples that are based on negation-normal-form. The following definition is used in ([Hyland, Sec 5.3](#Hyland)):

* A collection of objects, together with an involution operation $(-)^*$ (so $A^{**}=A$).
* For lists $\Gamma$, $\Delta$ of objects, a set $Hom(\Gamma;\Delta)$.
* _Exchange and negation structure:_ Two lists $\Gamma$ and $\Delta$ induce a single list $\Gamma^*,\Delta$ by concatenating $\Gamma^*$ with $\Delta$. Each object-preserving-bijection $\Gamma^*,\Delta\to\Pi^*,\Sigma$ should induce a function $Hom(\Gamma;\Delta)\to \Hom(\Pi;\Sigma)$, compatible with composition of bijections.
* For each object $A$, an identity morphism $Hom(A;A)$.
* For lists $\Gamma,\Delta,\Pi,\Sigma$ and objects $A$, a composition function
$$Hom(\Gamma;A,\Delta)\times Hom(\Pi,A;\Sigma)\to Hom(\Gamma,\Pi;\Delta,\Sigma)$$
respecting the exchange and negation structure, and satisfying the same identity and associativity laws as for an ordinary polycategory.

In the last axiom, "respecting the exchange and negation structure" has two pieces.  The first, more obvious, one is that the composition functions are natural with respect to object-preserving bijections $\Gamma_1^*,\Delta_1 \to \Gamma_2^*,\Delta_2$ and $\Pi_1^*,\Sigma_1 \to \Pi_2^*,\Sigma_2$ which leave the $A$'s untouched.  The second, perhaps less obvious, one is that composing along $A$ is the same as composing along $A^*$, in that the following diagram commutes:

$$
\array{
Hom(\Gamma;A,\Delta) \times Hom(\Pi,A;\Sigma) &\xrightarrow{\circ_A} & Hom(\Gamma,\Pi;\Delta,\Sigma) \\
\downarrow && \downarrow \\
Hom(\Gamma,A^*;\Delta) \times Hom(\Pi;A^*,\Sigma) && \downarrow \\
\downarrow && \downarrow \\
Hom(\Pi;A^*,\Sigma) \times Hom(\Gamma,A^*;\Delta) & \xrightarrow{\circ_{A^*}} & Hom(\Pi,\Gamma;\Sigma,\Delta)
}
$$

Note that because of symmetry, we can get away with requiring composition only when $A$ is the first object in $A,\Delta$ and the last object in $\Pi,A$.  If we included composition operations along an object $A$ with arbitrary placement in these lists, then there would be an axiom asserting that these compositions are invariant with respect to permutations that rearrange the location of $A$, hence essentially reducing them to the special case of a fixed placement (such as "first in $A,\Delta$ and last in $\Pi,A$").

### Entries only

In a star-polycategory we have natural bijections $Hom(\Gamma;\Delta)\cong Hom(-;\Gamma^*,\Delta)\cong Hom(\Gamma,\Delta^*;-)$. In many concrete examples these bijections are equalities. Moreover, it could be argued that in many concrete examples the explicit direction of morphisms in a two-sided polycategory isn't meaningful. This leads us to the following one-sided or entries only definition. It corresponds to one-sided presentation of sequent calculus.

* A collection of objects, together with an involution $(-)^*$ (so $A^{**}=A$).
* For any list $\Gamma$ of objects, a set $Hom(\Gamma)$ of "morphisms".
* Each object-preserving-bijection $\Gamma\to \Delta$ should induce a function $Hom(\Gamma)\to \Hom(\Delta)$, compatible with composition of bijections.
* For each object $A$, an identity morphism $Hom(A,A^*)$.
* For lists $\Gamma,\Delta$ and objects $A$, a composition function
$$Hom(\Gamma,A)\times Hom(A^*,\Delta)\to Hom(\Gamma,\Delta)$$
respecting the exchange structure and satisfying identity and associativity laws.

## Examples

* Recall that a [[linearly distributive category]] can be understood as a representable polycategory. It is star-autonomous if the corresponding polycategory is a star-polycategory.

* For example, [[profunctors]] form a (weak) star-polycategory. The objects are categories, $A^*=A^{\mathrm{op}}$, and (using the entries-only definition) $Hom(A_1,\dots,A_n)=Cat(A_1\dots\times\dots A_n,\mathbf{Set})$. The hom-functors are identities, and composition is given by [[coends]].

* The unit interval forms a thin star polycategory. The objects are real numbers in the interval $[0,1]$, and the involution is $A^*=(1-A)$. $Hom(A_1,\dots A_n)$ is inhabited if $\sum_{i=1}^n A_i = 1$, i.e. if the list partitions the interval. Note that this polycategory is not representable, for example, we cannot combine $1$ with anything except $0$.

* More generally, any [[effect algebra]] forms a thin star-polycategory in the same way. Sequences $\Gamma$ for which $Hom(\Gamma)$ is inhabited are usually called "tests". The tensor in the polycategorical sense is the partial monoid in the effect algebra sense.

## References

* [[Martin Hyland]], _Proof theory in the abstract_ ([pdf](https://www.dpmms.cam.ac.uk/~martin/Research/Publications/2002/pta02.pdf))
 {#Hyland}

[[!redirects star-polycategories]]
[[!redirects *-polycategories]]
[[!redirects *-polycategory]]


star-representation

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
#### Operator algebra
+-- {: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For $(\mathcal{A}, \ast)$ a [[star-algebra]] and $\mathcal{H}$ a [[Hilbert space]], a _star-representation_ of $(\mathcal{A},\ast)$ on $\mathcal{H}$ is a [[homomorphism]] of star algebras from $\mathcal{A}$ to [[linear operators]] on (a [[dense subspace]] of) $\mathcal{H}$.

## Related concepts

* [[representation of a C-star-algebra]]

* [[GNS construction]]

## References

* {#KhavkineMoretti15}  [[Igor Khavkine]], [[Valter Moretti]], def. 7 in _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

[[!redirects star-representations]]

[[!redirects *-representation]]
[[!redirects *-representations]]


Starobinsky model of cosmic inflation

+-- {: .rightHandSide}
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###Context###
#### Cosmology
+-- {: .hide}
[[!include cosmology -- contents]]
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#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

In [[phenomenology]] of [[cosmology]], the _Starobinsky model_ of [[cosmic inflation]] takes into account -- and takes as the very source of the [[inflaton]] field -- [[higher curvature corrections]] to the [[Einstein-Hilbert action]] of [[gravity]], notably the term $R^2$ (square of the [[Ricci curvature]]).

The Starobinsky model stands out among models of inflation as predicting a low value of the scalar-to-tensor ratio $r$, specifically it predicts

$$
  r \sim \frac{12}{N^2}
$$

where $N$ is the number of $e$-foldings during inflation (see e.g. [Kehagias-Dizgah-Riotto 13 (2.6)](#KehagiasDizgahRiotto13)).

## Observational support
 {#ObservationalSupport}

Models of Starobinsky-type are favored by [[experiment|experimental]] results ([PlanckCollaboration 13](#PlanckCollaboration13), [BICEP2-Keck-Planck 15](#BICEPKeckPlanck15), [PlanckCollaboration 15](#PlanckCollaboration15), [BICEP3-Keck 18](#BICEPKeck21)) which give a low upper bound on $r$, well below $0.1$ (whereas other models like [[chaotic inflation]] are disfavored by these values), see ([PlanckCollaboration 13, page 12](#PlanckCollaboration13)).

{#Dat} With respect to this data, the Starobinsky model (or "$R^2$ inflation") is the model with the highest [[Bayesian reasoning|Bayesian evidence]] ([Rachen, Feb 15](#Rachen15), [PlanckCollaboration 15XX, table 6 on p. 18](#PlanckCollaboration15XX)) as it is right in the center of the likelihood peak, shown in dark blue in the following plots ([PlanckCollaboration 13, figure 1](#PlanckCollaboration13), also [Linde 14, figure 5](#Linde14)) and at the same time has the lowest number of free parameters :

{#Figure1}

This remains true with the data of ([PlanckCollaboration 15](#PlanckCollaboration15)), see ([PlanckCollaboration 15 XIII, figure 22](#PlanckCollaboration15XIII)) and in the final analysis ([PlanckCollaboration 18X, Fig 8](#PlanckCollaboration18X)), which gives the following (from [here](http://resonaances.blogspot.se/2015/02/weekend-plot-inflation15.html)):

{#Figure2}

> $R^2$ inflation has the strongest evidence among the models considered here. However, care must be taken not to overinterpret small differences in likelihood lacking statistical significance. The models closest to $R^2$ in terms of evidence are brane inflation and exponential inflation, which have one more parameter
than $R^2$ ([PlanckCollaboration 15XX, p. 18](#PlanckCollaboration15XX))

{#KECKBICEPFurtherConfirmed} This picture is further confirmed by observations of the BICEP/Keck collaboration reported in [BICEP-Keck 2021](#BICEPKeck21), whose additional data singles out the dark blue area in the following (Fig. 5):

See also [Ellis 13](#Ellis13), [Ketov 13](#Ketov13), [Efstathiou 2019, 50:49](#Efstathiou2019) for brief survey of Starobinsky inflation in relation to observation, and see [Kehagias-Dizgah-Riotto 13](#KehagiasDizgahRiotto13) for more details. There it is argued that the other types of inflationary models which also reasonably fit the data are actually equivalent to the Starobinsky model during inflation.

## Embedding into supergravity
 {#EmbeddingIntoSupergravity}

Being concerned with pure [[gravity]] (the [[inflaton]] not being an extra [[matter]] field but part of the field of [[gravity]]) the Starobinsky model lends itself to embedding into [[supergravity]] (originally due to [Cecotti 87](#Cecotti87), see e.g. [Farakos-Kehagias-Riotto 13](#FKR13)). Such embedding has been argued to improve the model further (highlighted e.g. in [Ellis 13](#Ellis13)), for instance by

* {#ShrinkingOfInitialHomogeneousPatch} shrinking the necessary initial homogeneous patch from $\sim 10^3$ [[Planck lengths]] (which would be in need of further explanation) down to just $\sim 10^1$ [[Planck lengths]] ([Dalianis-Farakos 15 equations (68), (72) in v1, equations (4.11), (4.17) in v3](#DalianisFarakos15), reviewed in [Dalianis 16](#Dalianis16));

* {#CompatibilityWithsusybreaking} naturally subsuming a mechanism for [[supersymmetry breaking]] ([Ferrar-Kehagias 14](#FerrarKehagias14), [DFKRU 14](#DFKRU14)) notably with a Starobisnky potential naturally induced from [[gravitino condensation]] ([Alexandre-Houston-Mavromatos 14](#AlexandreHoustonMavromatos14)).

> graphics grabbed from [Dalianis 16, p. 8](#Dalianis16)

{#HCCIn11dSugra} More concretely, in [Hiraga-Hyakutake 18](#HiragaHyakutake18) a simple model of [[11-dimensional supergravity]] with its $R^4$ [[higher curvature correction]] (see [there](11-dimensional+supergravity#HigherCurvatureCorrection)) is considered and claimed to yield inflation with "graceful exit" and dynamical [[KK-compactification]]:

> graphics from [Hiraga-Hyakutake 18, p. 8](#HiragaHyakutake18)

## References

### General

The model is due to

* {#Starobisnky80} [[Aleksei Starobinsky]], *A new type of isotropic cosmological models without singularity*, Phys. Lett. B **91**  (1980) 99-102 \[doi:10.1016/0370-2693(80)90670-X\]

and the analysis of its predictions is due to:

* [[Viatcheslav Mukhanov]] and G. V. Chibisov, JETP Lett. **33** (1981) 532 and: Pisma Zh. Eksp. Teor. Fiz. **33**  (1981) 549 [[spire:170051](https://inspirehep.net/literature/170051)]

* [[Aleksei Starobinsky]], *Isotropization of arbitrary cosmological expansion given an effective cosmological constant*, JETP Lett. **37** (1983) 66-69 [[spire:187801](https://inspirehep.net/literature/187801)]

* [[Aleksei Starobinsky]], *The Perturbation Spectrum Evolving from a Nonsingular Initially De-Sitter Cosmology and the Microwave Background Anisotropy*, Sov. Astron. Lett. **9** (1983) 302 [[spire:199078](https://inspirehep.net/literature/199078)]

The experimental data supporting the model is due to

* {#PlanckCollaboration13} [[Planck Collaboration]], _Planck 2013 results. XXII. Constraints on inflation_ ([arXiv:1303.5082](http://arxiv.org/abs/1303.5082))

* {#BICEPKeckPlanck15} [[Planck Collaboration]], [[BICEP2]] _A Joint Analysis of BICEP2/Keck Array and Planck Data_ ([arXiv:1502.00612](http://arxiv.org/abs/1502.00612))

* {#PlanckCollaboration15} [[Planck Collaboration]], _Planck 2015, Overview of results_ ([arXiv:1502.01582](http://xxx.lanl.gov/abs/1502.01582))

* {#PlanckCollaboration15XIII} [[Planck Collaboration]], _Planck 2015 results. XIII. Cosmological parameters_ ([arXiv:1502.01589](http://xxx.lanl.gov/abs/1502.01589))

* {#PlanckCollaboration15XX} [[Planck Collaboration]], _Planck 2015 results. XX. Constraints on inflation_ ([arXiv:1502.02114](http://arxiv.org/abs/1502.02114))

* {#PlanckCollaboration18X} [[Planck Collaboration]], _Planck 2018 results. X. Constraints on inflation_ ([arXiv:1807.06211](https://arxiv.org/abs/1807.06211))

* {#BICEPKeck21} BICEP/Keck Collaboration: *BICEP / Keck XIII: Improved Constraints on Primordial Gravitational Waves using Planck, WMAP, and BICEP/Keck Observations through the 2018 Observing Season*, Phys. Rev. Lett. 127, 151301 ([arXiv:2110.00483](https://arxiv.org/abs/2110.00483), [doi:10.1103/PhysRevLett.127.151301](https://doi.org/10.1103/PhysRevLett.127.151301))

See also

* Debika Chowdhury, Jerome Martin, Christophe Ringeval, Vincent Vennin, _Inflation after Planck: Judgment Day_ ([arxiv:1902.03951](https://arxiv.org/abs/1902.03951))

* Swagat S. Mishra, Varun Sahni, *Canonical and Non-canonical Inflation in the light of the recent BICEP/Keck results* [[arXiv:2202.03467](https://arxiv.org/abs/2202.03467)]

* M. Chaichian, A. Ghal'e, M. Oksanen, *An alternative approach to the Starobinsky model for inflation scenario* [[arXiv:2211.05212](https://arxiv.org/abs/2211.05212)]

Review and exposition includes

* {#KehagiasDizgahRiotto13} [[Alex Kehagias]], Azadeh Moradinezhad Dizgah, Antonio Riotto, _Comments on the Starobinsky Model of Inflation and its Descendants_, Phys. Rev. D 89, 043527 (2014) ([arXiv:1312.1155](http://arxiv.org/abs/1312.1155))

* {#Ketov13} [[Sergei Ketov]], _PLANCK mission, Starobinsky inflation and its realization in old-minimal supergravity_, talk at _Kavli IPMU Workshop: SUSY Model Building and Phenomenology_, 2-4 December 2013 ([pdf](http://indico.ipmu.jp/indico/getFile.py/access?contribId=16&sessionId=7&resId=0&materialId=slides&confId=28))

* {#Ellis13} [[John Ellis]], _Planck-Compatible Inflationary Models_, talk 2013 ([pptx](http://londoncosmology.files.wordpress.com/2013/09/lcdm.pptx))

* {#Linde14} [[Andrei Linde]], _Inflationary Cosmology after Planck 2013_ ([arXiv:1402.0526](http://arxiv.org/abs/1402.0526))

* {#Rachen15} [[Jörg Rachen]], _The Planck 2015 Results: Cosmology and Fundamental Physics from the Polarised CMB and Other Probes_, IMAPP Special Seminar, Nijmegen, Feb.5, 2015

* {#Dalianis16} [[Ioannis Dalianis]],  _Features and implications of the plateau inflationary potentials_, Planck 2015 conference contribution ([arXiv:1602.05026](http://arxiv.org/abs/1602.05026))

* {#Efstathiou2019} George P. Efstathiou on behalf of the PLANCK mission, _The PLANCK legacy, inflation and the origin of structure in the universe_, talk at University of Cambridge, January 28, 2019 ([recording from 50:49](https://www.youtube.com/watch?v=16CkVzVK7Wk&feature=youtu.be&t=3049))

Discussion with more general [[higher curvature corrections]]:

* {#AEJ18} Gustavo Arciniega, Jose D. Edelstein, Luisa G. Jaime, _Towards purely geometric inflation and late time acceleration_ ([arXiv:1810.08166](https://arxiv.org/abs/1810.08166))

* Gustavo Arciniega, Pablo Bueno, Pablo A. Cano, Jose D. Edelstein, Robie A. Hennigar, Luisa G. Jaimem, _Geometric Inflation_ ([arXiv:1812.11187](https://arxiv.org/abs/1812.11187))

Discussion of [[eternal inflation]] in Starobinsky-type models

* {#BarenboimKinneyPark16} Gabriela Barenboim, [[William Kinney]], Wan-Il Park, _Eternal Hilltop Inflation_,  Journal of Cosmology and Astroparticle Physics, Volume 2016, May 2016 ([arXiv:1601.08140](https://arxiv.org/abs/1601.08140))

See also:

* Dhong Yeon Cheong, Hyun Min Lee, Seong Chan Park, _Beyond the Starobinsky model for inflation_ ([arXiv:2002.07981](https://arxiv.org/abs/2002.07981))

* Didam Duniya, Amare Abebe, Alvaro de la Cruz-Dombriz, Peter Dunsby, *Imprint of $f(R)$ gravity in the cosmic magnification* [[arXiv:2210.09303](https://arxiv.org/abs/2210.09303)]

### Embedding into supergravity
 {#ReferencesEmbeddingIntoSupergravity}

Discussion of embedding of Starobinsky inflation in [[supergravity]] originates in

* {#Cecotti87} [[Sergio Cecotti]], *Higher derivative supergravity is equivalent to standard supergravity coupled to matter*, Phys. Lett. B **190** 86 (1987) [doi:10.1016/0370-2693(87)90844-6]

* S. Cecotti, [[Sergio Ferrara]], M. Porrati and S. Sabharwal, Nucl. Phys. B 306, 160 (1988).

and is further developed in the following articles:

* [[Sergei Ketov]], _Supergravity and Early Universe: the Meeting Point of Cosmology and High-Energy Physics_, Int.J.Mod.Phys. A28 (2013) 1330021 ([arXiv:1201.2239](http://arxiv.org/abs/arXiv:1201.2239))

* [[John Ellis]], [[Dimitri Nanopoulos]], [[Keith Olive]], _No-Scale Supergravity Realization of the Starobinsky Model of Inflation_ Phys. Rev. Lett. 111, 111301 (2013) ([arXiv:1305.1247](http://arxiv.org/abs/1305.1247))

* [[Renata Kallosh]], [[Andrei Linde]], _Superconformal generalizations of the Starobinsky model_ JCAP 1306, 028 (2013) ([arXiv:1306.3214](http://arxiv.org/abs/1306.3214))

* [[John Ellis]], [[Dimitri Nanopoulos]], [[Keith Olive]], _Starobinsky-like Inflationary Models as Avatars of No-Scale Supergravity_ JCAP 1310, 009 (2013) ([arXiv:1307.3537](http://arxiv.org/abs/1307.3537))

* {#FKR13} [[Fotis Farakos]], [[Alex Kehagias]], A. Riotto, _On the Starobinsky Model of Inflation from Supergravity_, Nucl. Phys. B 876, 187 (2013) ([arXiv:1307.1137](http://arxiv.org/abs/1307.1137))

* [[Sergio Ferrara]], [[Renata Kallosh]], [[Andrei Linde]] and M. Porrati, _Minimal Supergravity Models of Inflation_ ([arXiv:1307.7696](http://arxiv.org/abs/1307.7696))

* [[Andrei Linde]] and M. Porrati, _Higher Order Corrections in Minimal Supergravity Models of Inflation_ ([arXiv:1309.1085](http://arxiv.org/abs/1309.1085))

* [[Sergio Ferrara]], [[Renata Kallosh]], [[Antoine Van Proeyen]], _On the Supersymmetric Completion of R+R2 Gravity and Cosmology_ JHEP 1311, 134 (2013) ([arXiv:1309.4052](http://arxiv.org/abs/1309.4052))

* [[Sergei Ketov]], Takahiro Terada, _Old-minimal supergravity models of inflation_, JHEP12(2013)040 ([arXiv:1309.7494](https://arxiv.org/abs/1309.7494))

* [[Sergio Ferrara]]], [[Pietro Fre]] and A. S. Sorin, _On the Topology of the Inflaton Field in Minimal Supergravity Models_ ([arXiv:1311.5059](http://arxiv.org/abs/1311.5059))

* {#AlexandreHoustonMavromatos14} Jean Alexandre, Nick Houston, Nick E. Mavromatos, _Starobinsky-type Inflation in Dynamical Supergravity Breaking Scenarios_, Phys. Rev. D 89, 027703 (2014) ([arXiv:1312.5197](https://arxiv.org/abs/1312.5197))

  via [[gravitino condensation]], based on

  * Jean Alexandre, Nick Houston, Nick E. Mavromatos, _Dynamical Supergravity Breaking via the Super-Higgs Effect Revisited_, Phys. Rev. D 88, 125017 (2013) ([arXiv:1310.4122](https://arxiv.org/abs/1310.4122))

  * Jean Alexandre, Nick Houston, Nick E. Mavromatos, _Inflation via Gravitino Condensation in Dynamically Broken Supergravity_,  International Journal of Modern Physics D, Volume 24, Issue 04, April 2015 ([arXiv:1409.3183](https://arxiv.org/abs/1409.3183))

* [[Sergei Ketov]], [[Aleksei Starobinsky]], _Inflation and non-minimal scalar-curvature coupling in gravity and supergravity_, JCAP 1208, 022 (2012) ([arXiv:1203.0805](https://arxiv.org/abs/1203.0805))

* [[Sergei Ketov]], S. Tsujikawa, _Consistency of inflation and preheating in $F(R)$ supergravity_, Phys. Rev. D 86, 023529 (2012) ([arXiv:1205.2918](https://arxiv.org/abs/1205.2918))

* [[Sergei Ketov]], _On the supersymmetrization of inflation in $f(R)$ gravity_,Prog. Theor. Exp. Phys. 2013, 123B04 ([arXiv:1309.0293](https://arxiv.org/abs/1309.0293))

* [[Sergio Ferrara]], [[Alex Kehagias]], Antonio Riotto, _The Imaginary Starobinsky Model and Higher Curvature Corrections_ ([arXiv:1405.2353](http://arxiv.org/abs/1405.2353))

* {#FerrarKehagias14} [[Sergio Ferrara]], [[Alex Kehagias]], _Higher Curvature Supergravity, Supersymmetry Breaking and Inflation_ ([arXiv:1407.5187](http://arxiv.org/abs/1407.5187))

* {#DFKRU14} [[Ioannis Dalianis]], [[Fotis Farakos]], [[Alex Kehagias]], A. Riotto, [[Rikard von Unge]], _Supersymmetry Breaking and Inflation from Higher Curvature Supergravity_ ([arXiv:1409.8299](http://arxiv.org/abs/1409.8299))

* {#DalianisFarakos15} [[Ioannis Dalianis]], [[Fotis Farakos]], _On the initial conditions for inflation with plateau potentials: the $R + R^2$ (super)gravity case_, [Journal of Cosmology and Astroparticle Physics, Volume 2015, July 2015 ](http://iopscience.iop.org/article/10.1088/1475-7516/2015/07/044/meta), ([arXiv:1502.01246](http://arxiv.org/abs/1502.01246))

* Spyros Basilakos, Nick E. Mavromatos, Joan Sola, _Starobinsky-like inflation and running vacuum in the context of Supergravity_ ([arXiv:1505.04434](https://arxiv.org/abs/1505.04434))

  > In this paper we have shown that SUGRA models with a dynamically induced massive gravitino phase lead to the RVM behavior and therefore provide a strong support for a fundamental description of the cosmic history.

* Andrea Addazi, [[Sergei Ketov]], _Energy conditions in Starobinsky supergravity_ ([arXiv:1701.02450](https://arxiv.org/abs/1701.02450))

* [[John Ellis]], Dimitri V. Nanopoulos, [[Keith Olive]], _From $R^2$ Gravity to No-Scale Supergravity_, Phys. Rev. D 97, 043530 (2018) ([arXiv:1711.11051](https://arxiv.org/abs/1711.11051))

* Hiroyuki Abe, Yermek Aldabergenov, Shuntaro Aoki, [[Sergei Ketov]], _Massive vector multiplet with Dirac-Born-Infeld and new Fayet-Iliopoulos terms in supergravity_ ([arXiv:1808.00669](https://arxiv.org/abs/1808.00669))

* Sergey Ketov, Maxim Khlopov, _Extending Starobinsky inflationary model in gravity and supergravity_ ([arXiv:1809.09975](https://arxiv.org/abs/1809.09975))

* {#ENOV19} [[John Ellis]], [[Dimitri Nanopoulos]], [[Keith Olive]], Sarunas Verner, _A Unified No-Scale Model of Modulus Fixing, Inflation, Supersymmetry Breaking and Dark Energy_, Phys. Rev. D 100, 025009 (2019) ([arXiv:1903.05267](https://arxiv.org/abs/1903.05267), [doi:10.1103/PhysRevD.100.025009](https://doi.org/10.1103/PhysRevD.100.025009))

* Yermek Aldabergenov, Shuntaro Aoki, [[Sergei Ketov]], _Minimal Starobinsky supergravity coupled to dilaton-axion superfield_ ([arXiv:2001.09574](https://arxiv.org/abs/2001.09574))

* [[John Ellis]], Marcos A. G. Garcia, Natsumi Nagata, [[Dimitri Nanopoulos]], [[Keith Olive]], Sarunas Verner, _Building Models of Inflation in No-Scale Supergravity_ ([arXiv:2009.01709](https://arxiv.org/abs/2009.01709))

* N. E. Martínez-Pérez, C. Ramírez, V. Vázquez-Báez, *FRW supersymmetric model of Starobinsky* ([arXiv:2110.00556](https://arxiv.org/abs/2110.00556))

### Embedding into 11d supergravity
 {#ReferencesEmbeddingInto11dSupergravity}

Discussion of Starobinsky inflation in [[11-dimensional supergravity]] with its [[higher curvature corrections]] included (see [there](11-dimensional+supergravity#HigherCurvatureCorrection)):

* [[Katrin Becker]], [[Melanie Becker]], _Supersymmetry Breaking, M-Theory and Fluxes_, JHEP 0107:038,2001 ([arXiv:hep-th/0107044](https://arxiv.org/abs/hep-th/0107044))

* {#HiragaHyakutake18} Kazuho Hiraga, Yoshifumi Hyakutake, _Inflationary Cosmology via Quantum Corrections in M-theory_ ([arXiv:1809.04724](https://arxiv.org/abs/1809.04724))

* Kazuho Hiraga, Yoshifumi Hyakutake, _Scalar Cosmological Perturbations in M-theory with Higher Derivative Corrections_ ([arxiv:1910.12483](https://arxiv.org/abs/1910.12483))

### Embedding into superstring theory
 {#ReferencesEmbeddingIntoStringTheory}

Embedding of Starobinsky inflation into [[superstring theory]] is discussed in

* [[Costas Kounnas]], [[Dieter Luest]], Nicolaos Toumbas, _$\mathcal{R}^2$ inflation from scale invariant supergravity and anomaly free superstrings with fluxes_ ([arXiv:1409.7076](http://arxiv.org/abs/1409.7076))

* [[Ralph Blumenhagen]], Anamaria Font, Michael Fuchs, Daniela Herschmann, Erik Plauschinn, _Towards Axionic Starobinsky-like Inflation in String Theory_, Physics Letters B Volume 746, 30 June 2015, Pages 217–222 ([arXiv:1503.01607](https://arxiv.org/abs/1503.01607))

* {#EllisCarciaNanopoulosOlive15} [[John Ellis]], Marcos A. G. Garcia, [[Dimitri Nanopoulos]], [[Keith Olive]], _Phenomenological Aspects of No-Scale Inflation Models_ ([arXiv:1503.08867](http://arxiv.org/abs/1503.08867))

* [[Luis Alvarez-Gaume]], [[Alex Kehagias]], [[Costas Kounnas]], [[Dieter Luest]], Antonio Riotto, _Aspects of Quadratic Gravity_ ([arXiv:1505.07657](http://arxiv.org/abs/1505.07657))

* Benedict Broy, David Ciupke, FranciscoG. Pedro, Alexander Westphal, _Starobinsky-Type Inflation from $\alpha'$-Corrections_, JCAP01(2016)001 ([arXiv:1509.00024](https://arxiv.org/abs/1509.00024))

* K. Sravan Kumar, _Inflaton candidates: from string theory to particle physics_, PhD thesis ([arXiv:1808.03701](https://arxiv.org/abs/1808.03701))

Concerning further [[higher curvature corrections]]:

* [[Sergei Ketov]], Ekaterina O. Pozdeeva, Sergey Yu. Vernov, *On the superstring-inspired quantum correction to the Starobinsky model of inflation* [[arXiv:2211.01546](https://arxiv.org/abs/2211.01546)]

On quasi-realistic "flipped" [[SU(5)|$SU(5)$]]-[[GUT]],  modeled in 4d [[heterotic string theory]] and subsuming realistic [[Starobinsky model of cosmic inflation|Starobinsky-type cosmic inflation]]:

* [[Ignatios Antoniadis]], [[John Ellis]], [[John S. Hagelin]], [[Dimitri Nanopoulos]], *The flipped $SU(5) \times U(1)$ string model revamped*, Physics Letters B **231** 1–2 (1989) 65-74 [doi:10.1016/0370-2693(89)90115-9]

* {#AntoniadisNanopoulosRizos21a} [[Ignatios Antoniadis]], [[Dimitri Nanopoulos]], [[John Rizos]], *Cosmology of the string derived flipped $SU(5)$*, J. Cosm. Astrop. Phys **03** (2021) 017 [[arXiv:2011.09396](https://arxiv.org/abs/2011.09396), [doi:10.1088/1475-7516/2021/03/017](https://doi.org/10.1088/1475-7516/2021/03/017)]

* {#AntoniadisNanopoulosRizos21a} [[Ignatios Antoniadis]], [[Dimitri Nanopoulos]], [[John Rizos]], *Particle physics and cosmology of the string derived no-scale flipped $SU(5)$*,  Eur. Phys. J. C **82** (2022) 377  [[arXiv:2112.01211](https://arxiv.org/abs/2112.01211), [doi:10.1140/epjc/s10052-022-10353-6](https://doi.org/10.1140/epjc/s10052-022-10353-6)]

[[!redirects Starobinsky inflation]]


state

+-- {: .rightHandSide}
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### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A _state_ of a system in [[physics]] is a description of everything that is true (or known, or relevant, depending on the circumstances) about the [[physical system]].

In the [[Bayesian interpretation of physics]], the state of a system is not a property of reality but instead indicates an observer\'s knowledge about the system.  A _[[pure state]]_ gives maximal information about the system (which amounts to complete information in [[classical mechanics]] but not generally in [[quantum mechanics]]), while a _[[mixed state]]_ is more general.  A mixed state can be decomposed into a [[probability distribution]] on the space of pure states, although this decomposition is unique only for classical systems.  In a frequentist interpretation of probability, a mixed state can describe only a statistical ensemble of systems; the real world is in one (generally unknown) pure state (possibly with additional hidden variables in the quantum case, depending on the interpretation of quantum physics).

States in the [[Schrödinger picture]] describe the state of the system at any given time and are subject to [[time evolution]], while in the [[Heisenberg picture]] a single state describes the entire history of the system (with time evolution applying to the [[observables]] instead).

In [[statistical physics]], a [[microstate]] is a complete description of all of the particles in a system, while a [[macrostate]] considers only the macroscopic properties known to [[thermodynamics]].

## Definitions

The precise mathematical notion of _state_ depends on what mathematical formalization of mechanics is used.

### Algebraic definition
 {#AlgebraicDefinition}

Quite generally, both in [[classical physics]] as well as in [[quantum physics]], one may define states as assignments of [[expectation values]] to [[observables]] in an [[algebra of observables]]. This is the definition used in _[[quantum probability theory]]_ (which subsumes ordinary [[probability theory]]). See at

* _[[state in AQFT and operator algebra]]_

for details.

### In classical mechanics

In classical [[Lagrangian mechanics]], a pure state is a [[global element|point]] in the [[state space]] of the system, giving all of the (generalised) [[generalized position|positions]] and [[velocity|velocities]].  In classical [[Hamiltonian mechanics]], a pure state is a point in the [[phase space]] of the system, giving the positions and [[momentum|momenta]].  In either case, a mixed state is a [[probability distribution]] on the space of pure states.

More generally, a _[[classical state]]_ is a [[linear function]] $\rho\colon A \to \mathbb{R}$ on the [[Poisson algebra]] $A$ underlying the [[classical mechanical system]] which satisfies _positivity_ and _normalization_.

### In geometric quantization

* [[space of states (in geometric quantization)]]

### In Hilbert-space quantum mechanics

In [[quantum mechanics]] given by a [[Hilbert space]] $H$, a [[pure state]] is a ray in $H$, which we often call the Hilbert space of states.  Strictly speaking, the space of states is not $H$ but $(H \setminus \{0\})/\mathbb{C}$, or equivalently $S(H)/\mathrm{U}(1)$.  A mixed state is then a [[density matrix]] on $H$.

### In AQFT

In [[AQFT]], a [[quantum mechanical system]] is given by a $C^*$-[[C-star-algebra|algebra]] $A$, and a [[quantum state]] is usually defined as a linear function $\rho\colon A \to \mathbb{C}$ which satisfies _positivity_ and _normalization_; see [[states in AQFT and operator algebra]].

Arguably, the correct notion of state to use is that of [[quasi-state]]; every state gives rise to a unique quasi-state, but not conversely.  However, when either classical mechanics or Hilbert-space quantum mechanics is formulated in AQFT, every quasi-state *is* a state (at least if the Hilbert space is not of very low dimension, by [[Gleason's theorem]]).  See also the Idea-section at [[Bohr topos]] for a discussion of this point.

### In FQFT

In the [[FQFT]] formulation of [[quantum field theory]], a physical system is given by a [[cobordism]] [[representation]]

$$
  Bord_n^S \to \mathcal{C}
  \,.
$$

In this formulation the [[k-morphism|(n-1)-morphism]] in $\mathcal{C}$ assigned to an $(n-1)$-dimensional [[manifold]] $\Sigma_{n-1}$ is the _space of states_ over that manifold. A state is accordingly a [[generalized element]] of this object.

## Pure and mixed states

In [[statistical physics]], a [[pure state]] is a state of maximal information, while a [[mixed state]] is a state with less than maximal information.  In the classical case, we may say that a pure state is a state of *complete* information, but this does not work in the quantum case; from the perspective of the information-theoretic or Bayesian interpretation of quantum physics, this inability to have complete information, even when having maximal information, is the key feature of [[quantum physics]] that distinguishes it from [[classical physics]].

See [[pure state]].

## Examples

Here are some toy examples of spaces of states.

For an impossible system, the space of states is [[empty space|empty]]; for a trivial system (with a unique way to be), the space of states is the [[point]].  This unique state is pure.

For a classical [[bit]], a system with two distinct ways to be, the space of states is a [[line segment]]; a state is given by a real number $t$ with $0 \leq t \leq 1$.  This $t$ is the probability that the system is in the first state, with $1 - t$ the probability that it is in the second; the two pure states correspond to $t = 0$ and $t = 1$.  This space naturally comes equipped with a metric structure in which its length is actually $2$; it's the line segment from $(0,1)$ to $(1,0)$ in the [[Lebesgue space]] $l^1(2)$.

For a quantum bit, a [[qubit]], a state is given by a self-adjoint $2$-by-$2$ complex matrix
$$ \begin{pmatrix} a & b - \mathrm{i} c \\ b + \mathrm{i} c & d \end{pmatrix} $$
with unit trace (so $d = 1 - a$) and nonnegative determinant (so $a^2 + b^2 + c^2 \leq a$).
However, the metric structure from viewing this as a subset of $\mathbb{R}^3$ is misleading, so we use $w \coloneqq a - d = 2 a - 1$ instead, along with $u \coloneqq 2 b$ and $v \coloneqq 2 c$ to keep the same normalization.  Then the density matrix may be written
$$ \frac 1 2 \begin{pmatrix} 1 + w & u - \mathrm{i} v \\ u + \mathrm{i} v & 1 - w \end{pmatrix} = \frac 1 2 (\mathrm{I} + u \sigma_x + v \sigma_y + w \sigma_z) ,$$
where the $\sigma$s are the [[Pauli matrices]].  This already has unit trace; the inequality for the determinant is
$$ u^2 + v^2 + w^2 \leq 1 .$$
The pure states are those satisfying $u^2 + v^2 + w^2 = 1$, forming a unit [[sphere]], known in this context as the [[Bloch sphere]] (so the full space of all states is the Bloch ball).

## Related concepts

[[!include states and observables -- content]]

\linebreak

[[!include Isbell duality - table]]

[[!redirects state]]
[[!redirects states]]
[[!redirects physical state]]
[[!redirects physical states]]

[[!redirects space of states]]
[[!redirects spaces of states]]

[[!redirects space of physical states]]
[[!redirects spaces of physicsal states]]