Wigner–Weyl transform

<h2 id="definition-of-the-weyl-quantization-of-a-general-observable">Definition of the Weyl quantization of a general observable</h2>
The following explains the Weyl transformation on the simplest, two-dimensional Euclidean phase space. Let the coordinates on phase space be (q,p), and let f be a function defined everywhere on phase space. In what follows, we fix operators P and Q satisfying the <a href="/facts/Canonical_commutation_relation/UgUW6TuT">canonical commutation relations</a>, such as the usual position and momentum operators in the Schrödinger representation. We assume that the exponentiated operators 
 
 
 
 
 e
 
 i
 a
 Q
 
 
 
 
 {\displaystyle e^{iaQ}}
 
 and 
 
 
 
 
 e
 
 i
 b
 P
 
 
 
 
 {\displaystyle e^{ibP}}
 
 constitute an irreducible representation of the <a href="/facts/Stone%25E2%2580%2593von_Neumann_theorem/4C64mKYR">Weyl relations</a>, so that the <a href="/facts/Stone%25E2%2580%2593von_Neumann_theorem/4C64mKYR">Stone–von Neumann theorem</a> (guaranteeing uniqueness of the canonical commutation relations) holds. 

<h3>Basic formula</h3>
The Weyl transform (or Weyl quantization) of the function f is given by the following operator in Hilbert space,<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a><a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

Φ
 [
 f
 ]
 =
 
 
 1
 
 (
 2
 π
 
 )
 
 2
 
 
 
 
 
 ∬
 
 
 
 ∬
 f
 (
 q
 ,
 p
 )
 
 (
 
 e
 
 i
 (
 a
 (
 Q
 −
 q
 )
 +
 b
 (
 P
 −
 p
 )
 )
 
 
 )
 
 
 d
 
 q
 
 
 d
 
 p
 
 
 d
 
 a
 
 
 d
 
 b
 .
 
 
 {\displaystyle \Phi [f]={\frac {1}{(2\pi )^{2}}}\iint \!\!\!\iint f(q,p)\left(e^{i(a(Q-q)+b(P-p))}\right){\text{d}}q\,{\text{d}}p\,{\text{d}}a\,{\text{d}}b.}

Throughout, ħ is the <a href="/facts/Reduced_Planck_constant/qGHAe3qq">reduced Planck constant</a>. 
It is instructive to perform the p and q integrals in the above formula first, which has the effect of computing the ordinary Fourier transform 
 
 
 
 
 
 
 f
 ~
 
 
 
 
 
 {\displaystyle {\tilde {f}}}
 
 of the function f, while leaving the operator 
 
 
 
 
 e
 
 i
 (
 a
 Q
 +
 b
 P
 )
 
 
 
 
 {\displaystyle e^{i(aQ+bP)}}
 
. In that case, the Weyl transform can be written as<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

Φ
 [
 f
 ]
 =
 
 
 1
 
 (
 2
 π
 
 )
 
 2
 
 
 
 
 
 ∬
 
 
 
 f
 ~
 
 
 
 (
 a
 ,
 b
 )
 
 e
 
 i
 a
 Q
 +
 i
 b
 P
 
 
 
 d
 a
 
 d
 b
 
 
 {\displaystyle \Phi [f]={\frac {1}{(2\pi )^{2}}}\iint {\tilde {f}}(a,b)e^{iaQ+ibP}\,da\,db}
 
.
We may therefore think of the Weyl map as follows: We take the ordinary Fourier transform of the function 
 
 
 
 f
 (
 p
 ,
 q
 )
 
 
 {\displaystyle f(p,q)}
 
, but then when applying the Fourier inversion formula, we substitute the quantum operators 
 
 
 
 P
 
 
 {\displaystyle P}
 
 and 
 
 
 
 Q
 
 
 {\displaystyle Q}
 
 for the original classical variables p and q, thus obtaining a "quantum version of f."
A less symmetric form, but handy for applications, is the following,

Φ
        [
        f
        ]
        =
        
          
            2
            
              (
              2
              π
              ℏ
              
                )
                
                  3
                  
                    /
                  
                  2
                
              
            
          
        
        ∬
        
        
        
        ∬
        
        
        d
        q
        
        d
        p
        
        d
        
          
            
              x
              ~
            
          
        
        
        d
        
          
            
              p
              ~
            
          
        
         
        
          e
          
            
              
                i
                ℏ
              
            
            (
            
              
                
                  x
                  ~
                
              
            
            
              
                
                  p
                  ~
                
              
            
            −
            2
            (
            
              
                
                  p
                  ~
                
              
            
            −
            p
            )
            (
            
              
                
                  x
                  ~
                
              
            
            −
            q
            )
            )
          
        
         
        f
        (
        q
        ,
        p
        )
         
        
          |
        
        
          
            
              x
              ~
            
          
        
        ⟩
        ⟨
        
          
            
              p
              ~
            
          
        
        
          |
        
        .
      
    
    {\displaystyle \Phi [f]={\frac {2}{(2\pi \hbar )^{3/2}}}\iint \!\!\!\iint \!\!dq\,dp\,d{\tilde {x}}\,d{\tilde {p}}\ e^{{\frac {i}{\hbar }}({\tilde {x}}{\tilde {p}}-2({\tilde {p}}-p)({\tilde {x}}-q))}~f(q,p)~|{\tilde {x}}\rangle \langle {\tilde {p}}|.}

<h3>In the position representation</h3>
The Weyl map may then also be expressed in terms of the integral kernel matrix elements of this operator,<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a>

⟨
        x
        
          |
        
        Φ
        [
        f
        ]
        
          |
        
        y
        ⟩
        =
        
          ∫
          
            −
            ∞
          
          
            ∞
          
        
        
          
            
              
                d
              
              p
            
            h
          
        
         
        
          e
          
            i
            p
            (
            x
            −
            y
            )
            
              /
            
            ℏ
          
        
         
        f
        
          (
          
            
              
                
                  x
                  +
                  y
                
                2
              
            
            ,
            p
          
          )
        
        .
      
    
    {\displaystyle \langle x|\Phi [f]|y\rangle =\int _{-\infty }^{\infty }{{\text{d}}p \over h}~e^{ip(x-y)/\hbar }~f\left({x+y \over 2},p\right).}

<h3>Inverse map</h3>
The inverse of the above Weyl map is the Wigner map (or Wigner transform), which was introduced by Eugene Wigner,<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> which takes the operator Φ back to the original phase-space kernel function f,

f
 (
 q
 ,
 p
 )
 =
 2
 
 ∫
 
 −
 ∞
 
 
 ∞
 
 
 
 d
 
 y
  
 
 e
 
 −
 2
 i
 p
 y
 
 /
 
 ℏ
 
 
  
 ⟨
 q
 +
 y
 
 |
 
 Φ
 [
 f
 ]
 
 |
 
 q
 −
 y
 ⟩
 .
 
 
 {\displaystyle f(q,p)=2\int _{-\infty }^{\infty }{\text{d}}y~e^{-2ipy/\hbar }~\langle q+y|\Phi [f]|q-y\rangle .}

For example, the Wigner map of the oscillator thermal distribution operator 
 
 
 
 exp
 ⁡
 (
 −
 β
 (
 
 P
 
 2
 
 
 +
 
 Q
 
 2
 
 
 )
 
 /
 
 2
 )
 
 
 {\displaystyle \exp(-\beta (P^{2}+Q^{2})/2)}
 
 is<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>

exp
          
            ⋆
          
        
        ⁡
        
          (
          
            −
            β
            (
            
              p
              
                2
              
            
            +
            
              q
              
                2
              
            
            )
            
              /
            
            2
          
          )
        
        =
        
          
            (
            
              cosh
              ⁡
              
                (
                
                  
                    
                      β
                      ℏ
                    
                    2
                  
                
                )
              
            
            )
          
          
            −
            1
          
        
        exp
        ⁡
        
          (
          
            
              
                
                  −
                  2
                
                ℏ
              
            
            tanh
            ⁡
            
              (
              
                
                  
                    β
                    ℏ
                  
                  2
                
              
              )
            
            (
            
              p
              
                2
              
            
            +
            
              q
              
                2
              
            
            )
            
              /
            
            2
          
          )
        
        .
      
    
    {\displaystyle \exp _{\star }\left(-\beta (p^{2}+q^{2})/2\right)=\left(\cosh \left({\frac {\beta \hbar }{2}}\right)\right)^{-1}\exp \left({\frac {-2}{\hbar }}\tanh \left({\frac {\beta \hbar }{2}}\right)(p^{2}+q^{2})/2\right).}

If one replaces 
 
 
 
 Φ
 [
 f
 ]
 
 
 {\displaystyle \Phi [f]}
 
 in the above expression with an arbitrary operator, the resulting function f may depend on the reduced Planck constant ħ, and may well describe quantum-mechanical processes, provided it is properly composed through the <a href="/facts/Moyal_product/Dxn1R8cm">star product</a>, below.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>
In turn, the Weyl map of the Wigner map is summarized by Groenewold's formula,<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>

Φ
        [
        f
        ]
        =
        h
        ∬
        
        d
        a
        
        d
        b
         
        
          e
          
            i
            a
            Q
            +
            i
            b
            P
          
        
        Tr
        ⁡
        (
        
          e
          
            −
            i
            a
            Q
            −
            i
            b
            P
          
        
        Φ
        )
        .
      
    
    {\displaystyle \Phi [f]=h\iint \,da\,db~e^{iaQ+ibP}\operatorname {Tr} (e^{-iaQ-ibP}\Phi ).}

<h3>Weyl quantization of polynomial observables</h3>
While the above formulas give a nice understanding of the Weyl quantization of a very general observable on phase space, they are not very convenient for computing on simple observables, such as those that are polynomials in 
 
 
 
 q
 
 
 {\displaystyle q}
 
 and 
 
 
 
 p
 
 
 {\displaystyle p}
 
. In later sections, we will see that on such polynomials, the Weyl quantization represents the totally symmetric ordering of the noncommuting operators 
 
 
 
 Q
 
 
 {\displaystyle Q}
 
 and 
 
 
 
 P
 
 
 {\displaystyle P}
 
.
For example, the Wigner map of the quantum angular-momentum-squared operator L2 is not just the classical angular momentum squared, but it further contains an offset term −3ħ2/2, which accounts for the nonvanishing angular momentum of the ground-state <a href="/facts/Bohr_model/FmTHjaxy">Bohr orbit</a>.

<h2 id="properties">Properties</h2>
<h3>Weyl quantization of polynomials</h3>
The action of the Weyl quantization on polynomial functions of 
 
 
 
 q
 
 
 {\displaystyle q}
 
 and 
 
 
 
 p
 
 
 {\displaystyle p}
 
 is completely determined by the following symmetric formula:<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a>

(
        a
        q
        +
        b
        p
        
          )
          
            n
          
        
        ⟼
        (
        a
        Q
        +
        b
        P
        
          )
          
            n
          
        
      
    
    {\displaystyle (aq+bp)^{n}\longmapsto (aQ+bP)^{n}}

for all complex numbers 
 
 
 
 a
 
 
 {\displaystyle a}
 
 and 
 
 
 
 b
 
 
 {\displaystyle b}
 
. From this formula, it is not hard to show that the Weyl quantization on a function of the form 
 
 
 
 
 q
 
 k
 
 
 
 p
 
 l
 
 
 
 
 {\displaystyle q^{k}p^{l}}
 
 gives the average of all possible orderings of 
 
 
 
 k
 
 
 {\displaystyle k}
 
 factors of 
 
 
 
 Q
 
 
 {\displaystyle Q}
 
 and 
 
 
 
 l
 
 
 {\displaystyle l}
 
 factors of 
 
 
 
 P
 
 
 {\displaystyle P}
 
:
 
 
 
 
 ∏
 
 j
 =
 1
 
 
 N
 
 
 
 ξ
 
 
 k
 
 j
 
 
 
 
  
  
 ⟼
  
  
 
 
 1
 
 N
 !
 
 
 
 
 ∑
 
 σ
 ∈
 
 S
 
 N
 
 
 
 
 
 ∏
 
 j
 =
 1
 
 
 N
 
 
 
 Ξ
 
 
 k
 
 σ
 (
 j
 )
 
 
 
 
 
 
 {\displaystyle \prod _{j=1}^{N}\xi _{k_{j}}~~\longmapsto ~~{\frac {1}{N!}}\sum _{\sigma \in S_{N}}\prod _{j=1}^{N}\Xi _{k_{\sigma (j)}}}
 
where 
 
 
 
 
 ξ
 
 j
 
 
 =
 
 q
 
 j
 
 
 ,
 
 ξ
 
 n
 +
 j
 
 
 =
 
 p
 
 j
 
 
 
 
 {\displaystyle \xi _{j}=q_{j},\xi _{n+j}=p_{j}}
 
, and 
 
 
 
 
 S
 
 N
 
 
 
 
 {\displaystyle S_{N}}
 
 is the set of <a href="/facts/Permutation_group/vRsNNwwx">permutations on N elements</a>.
For example, we have

6
        
          p
          
            2
          
        
        
          q
          
            2
          
        
         
         
        ⟼
         
         
        
          P
          
            2
          
        
        
          Q
          
            2
          
        
        +
        
          Q
          
            2
          
        
        
          P
          
            2
          
        
        +
        P
        Q
        P
        Q
        +
        P
        
          Q
          
            2
          
        
        P
        +
        Q
        P
        Q
        P
        +
        Q
        
          P
          
            2
          
        
        Q
        .
      
    
    {\displaystyle 6p^{2}q^{2}~~\longmapsto ~~P^{2}Q^{2}+Q^{2}P^{2}+PQPQ+PQ^{2}P+QPQP+QP^{2}Q.}

While this result is conceptually natural, it is not convenient for computations when 
 
 
 
 k
 
 
 {\displaystyle k}
 
 and 
 
 
 
 l
 
 
 {\displaystyle l}
 
 are large. In such cases, we can use instead McCoy's formula<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a>

p
          
            m
          
        
        
          q
          
            n
          
        
         
         
        ⟼
         
         
        
          
            1
            
              2
              
                n
              
            
          
        
        
          ∑
          
            r
            =
            0
          
          
            n
          
        
        
          
            
              (
            
            
              n
              r
            
            
              )
            
          
        
        
          Q
          
            r
          
        
        
          P
          
            m
          
        
        
          Q
          
            n
            −
            r
          
        
        =
        
          
            1
            
              2
              
                m
              
            
          
        
        
          ∑
          
            s
            =
            0
          
          
            m
          
        
        
          
            
              (
            
            
              m
              s
            
            
              )
            
          
        
        
          P
          
            s
          
        
        
          Q
          
            n
          
        
        
          P
          
            m
            −
            s
          
        
        .
      
    
    {\displaystyle p^{m}q^{n}~~\longmapsto ~~{1 \over 2^{n}}\sum _{r=0}^{n}{n \choose r}Q^{r}P^{m}Q^{n-r}={1 \over 2^{m}}\sum _{s=0}^{m}{m \choose s}P^{s}Q^{n}P^{m-s}.}

This expression gives an apparently different answer for the case of 
 
 
 
 
 p
 
 2
 
 
 
 q
 
 2
 
 
 
 
 {\displaystyle p^{2}q^{2}}
 
 from the totally symmetric expression above. There is no contradiction, however, since the canonical commutation relations allow for more than one expression for the same operator. (The reader may find it instructive to use the commutation relations to rewrite the totally symmetric formula for the case of 
 
 
 
 
 p
 
 2
 
 
 
 q
 
 2
 
 
 
 
 {\displaystyle p^{2}q^{2}}
 
 in terms of the operators 
 
 
 
 
 P
 
 2
 
 
 
 Q
 
 2
 
 
 
 
 {\displaystyle P^{2}Q^{2}}
 
, 
 
 
 
 Q
 
 P
 
 2
 
 
 Q
 
 
 {\displaystyle QP^{2}Q}
 
, and 
 
 
 
 
 Q
 
 2
 
 
 
 P
 
 2
 
 
 
 
 {\displaystyle Q^{2}P^{2}}
 
 and verify the first expression in McCoy's formula with 
 
 
 
 m
 =
 n
 =
 2
 
 
 {\displaystyle m=n=2}
 
.)
It is widely thought that the Weyl quantization, among all quantization schemes, comes as close as possible to mapping the Poisson bracket on the classical side to the commutator on the quantum side. (An exact correspondence is impossible, in light of <a href="/facts/Canonical_quantization/AKHX2JQn">Groenewold's theorem</a>.) For example, Moyal showed the 

Theorem: If 
 
 
 
 f
 (
 q
 ,
 p
 )
 
 
 {\displaystyle f(q,p)}
 
 is a polynomial of degree at most 2 and 
 
 
 
 g
 (
 q
 ,
 p
 )
 
 
 {\displaystyle g(q,p)}
 
 is an arbitrary polynomial, then we have 
 
 
 
 Φ
 (
 {
 f
 ,
 g
 }
 )
 =
 
 
 1
 
 i
 ℏ
 
 
 
 [
 Φ
 (
 f
 )
 ,
 Φ
 (
 g
 )
 ]
 
 
 {\displaystyle \Phi (\{f,g\})={\frac {1}{i\hbar }}[\Phi (f),\Phi (g)]}
 
.
<h3>Weyl quantization of general functions</h3>
<ul><li>If f is a <a href="/facts/Real-valued_function/jetqlJpK">real-valued function</a>, then its Weyl-map image Φ[f] is <a href="/facts/Self-adjoint/rKqGtlpy">self-adjoint</a>.</li>
<li>If f is an element of <a href="/facts/Schwartz_space/B035faw7">Schwartz space</a>, then Φ[f] is <a href="/facts/Trace-class/2AmWENA3">trace-class</a>.</li>
<li>More generally, Φ[f] is a densely defined <a href="/facts/Unbounded_operator/5u2njqHt">unbounded operator</a>.</li>
<li>The map Φ[f] is one-to-one on the Schwartz space (as a subspace of the square-integrable functions).</li></ul>
<h2 id="see-also">See also</h2>

<ul><li><a href="/facts/Canonical_commutation_relation/UgUW6TuT">Canonical commutation relation</a></li>
<li><a href="/facts/Deformation_quantization/4lSVKfhv">Deformation quantization</a></li>
<li><a href="/facts/Heisenberg_group/rsqq1426">Heisenberg group</a></li>
<li><a href="/facts/Moyal_bracket/EtUA7phv">Moyal bracket</a></li>
<li><a href="/facts/Weyl_algebra/PADSD0k9">Weyl algebra</a></li>
<li><a href="/facts/Functor/l2u3rIBE">Functor</a></li>
<li><a href="/facts/Pseudo-differential_operator/wEPLVm5c">Pseudo-differential operator</a></li>
<li><a href="/facts/Wigner_quasi-probability_distribution/5tyVfdFF">Wigner quasi-probability distribution</a></li>
<li><a href="/facts/Stone%25E2%2580%2593von_Neumann_theorem/4C64mKYR">Stone–von Neumann theorem</a></li>
<li><a href="/facts/Phase_space_formulation/ogvtdMNs">Phase space formulation</a> of quantum mechanics</li>
<li><a href="/facts/Kontsevich_quantization_formula/HaISztzG">Kontsevich quantization formula </a></li>
<li><a href="/facts/Gabor%25E2%2580%2593Wigner_transform/4ErHlog2">Gabor–Wigner transform</a></li>
<li><a href="/facts/Oscillator_representation/k1G8R5qx">Oscillator representation</a></li></ul>

<ul><li>Hall, Brian C. (2013), Quantum Theory for Mathematicians, Graduate Texts in Mathematics, vol. 267, Springer, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2013qtm..book.....H">2013qtm..book.....H</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1461471158</li></ul>
<h2 id="further-reading">Further reading</h2>
<ul><li>Case, William B. (October 2008). "Wigner functions and Weyl transforms for pedestrians". <a href="/facts/American_Journal_of_Physics/GYzhvGaC">American Journal of Physics</a>. 76 (10): 937–946. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2008AmJPh..76..937C">2008AmJPh..76..937C</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1119%2F1.2957889">10.1119/1.2957889</a>. (Sections I to IV of this article provide an overview over the Wigner–Weyl transform, the <a href="/facts/Wigner_quasiprobability_distribution/5tyVfdFF">Wigner quasiprobability distribution</a>, the <a href="/facts/Phase_space_formulation/ogvtdMNs">phase space formulation</a> of quantum mechanics and the example of the <a href="/facts/Quantum_harmonic_oscillator/JYA5J0h5">quantum harmonic oscillator</a>.)</li>
<li><a href="https://www.encyclopediaofmath.org/index.php?title=Weyl_quantization">"Weyl quantization"</a>, <a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li>
<li>Terence Tao's 2012 <a href="https://terrytao.wordpress.com/2012/10/07/some-notes-on-weyl-quantisation/">notes on Weyl ordering</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Weyl, H. (1927). "Quantenmechanik und Gruppentheorie". Zeitschrift für Physik. 46 (1–2): 1–46. Bibcode:1927ZPhy...46....1W. doi:10.1007/BF02055756. S2CID 121036548. <a href="/wiki/Zeitschrift_f%C3%BCr_Physik" target="_blank">/wiki/Zeitschrift_f%C3%BCr_Physik</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Groenewold, H. J. (1946). "On the Principles of elementary quantum mechanics". Physica. 12 (7): 405–446. Bibcode:1946Phy....12..405G. doi:10.1016/S0031-8914(46)80059-4. <a href="/wiki/Physica_(journal)" target="_blank">/wiki/Physica_(journal)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Moyal, J. E.; Bartlett, M. S. (1949). "Quantum mechanics as a statistical theory". Mathematical Proceedings of the Cambridge Philosophical Society. 45 (1): 99–124. Bibcode:1949PCPS...45...99M. doi:10.1017/S0305004100000487. S2CID 124183640. <a href="/wiki/Jos%C3%A9_Enrique_Moyal" target="_blank">/wiki/Jos%C3%A9_Enrique_Moyal</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Curtright, T. L.; Zachos, C. K. (2012). "Quantum Mechanics in Phase Space". Asia Pacific Physics Newsletter. 1: 37–46. arXiv:1104.5269. doi:10.1142/S2251158X12000069. S2CID 119230734. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Folland, G. (1989). Harmonic Analysis in Phase Space. The Annals of mathematics studies. Vol. 122. Princeton, N.J.: Princeton University Press. ISBN 978-0-691-08528-9. <a href="978-0-691-08528-9" target="_blank">978-0-691-08528-9</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Curtright, T. L.; Fairlie, D. B.; Zachos, C. K. (2014). A Concise Treatise on Quantum Mechanics in Phase Space (PDF). World Scientific. ISBN 9789814520430. <a href="9789814520430" target="_blank">9789814520430</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Hall 2013 Section 13.3 - Hall, Brian C. (2013), Quantum Theory for Mathematicians, Graduate Texts in Mathematics, vol. 267, Springer, Bibcode:2013qtm..book.....H, ISBN 978-1461471158 <a href="https://ui.adsabs.harvard.edu/abs/2013qtm..book.....H" target="_blank">https://ui.adsabs.harvard.edu/abs/2013qtm..book.....H</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Hall 2013 Definition 13.7 - Hall, Brian C. (2013), Quantum Theory for Mathematicians, Graduate Texts in Mathematics, vol. 267, Springer, Bibcode:2013qtm..book.....H, ISBN 978-1461471158 <a href="https://ui.adsabs.harvard.edu/abs/2013qtm..book.....H" target="_blank">https://ui.adsabs.harvard.edu/abs/2013qtm..book.....H</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Wigner, E. (1932). "On the Quantum Correction for Thermodynamic Equilibrium". Physical Review. 40 (5): 749–759. doi:10.1103/PhysRev.40.749. <a href="/wiki/Physical_Review" target="_blank">/wiki/Physical_Review</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Curtright, T. L.; Fairlie, D. B.; Zachos, C. K. (2014). A Concise Treatise on Quantum Mechanics in Phase Space (PDF). World Scientific. ISBN 9789814520430. <a href="9789814520430" target="_blank">9789814520430</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Kubo, R. (1964). "Wigner Representation of Quantum Operators and Its Applications to Electrons in a Magnetic Field". Journal of the Physical Society of Japan. 19 (11): 2127–2139. Bibcode:1964JPSJ...19.2127K. doi:10.1143/JPSJ.19.2127. <a href="/wiki/Journal_of_the_Physical_Society_of_Japan" target="_blank">/wiki/Journal_of_the_Physical_Society_of_Japan</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Curtright, T. L.; Fairlie, D. B.; Zachos, C. K. (2014). A Concise Treatise on Quantum Mechanics in Phase Space (PDF). World Scientific. ISBN 9789814520430. <a href="9789814520430" target="_blank">9789814520430</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Hall 2013 Proposition 13.3 - Hall, Brian C. (2013), Quantum Theory for Mathematicians, Graduate Texts in Mathematics, vol. 267, Springer, Bibcode:2013qtm..book.....H, ISBN 978-1461471158 <a href="https://ui.adsabs.harvard.edu/abs/2013qtm..book.....H" target="_blank">https://ui.adsabs.harvard.edu/abs/2013qtm..book.....H</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">McCoy, Neal (1932). "On the Function in Quantum Mechanics which Corresponds to a Given Function in Classical Mechanics", Proc Nat Acad Sci USA 19 674, online . <a href="https://www.jstor.org/stable/85974?seq=1#page_scan_tab_contents" target="_blank">https://www.jstor.org/stable/85974?seq=1#page_scan_tab_contents</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
</ol>

Wigner–Weyl transform open-in-new

Wigner–Weyl transform