Stone's theorem on one-parameter unitary groups

<h2 id="formal-statement">Formal statement</h2>
<p>The statement of the theorem is as follows.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> 
</p>
Theorem. Let 
  
    
      
        (
        
          U
          
            t
          
        
        
          )
          
            t
            ∈
            
              R
            
          
        
      
    
    {\displaystyle (U_{t})_{t\in \mathbb {R} }}
  
 be a <a href="/facts/Compact_operator/KNkcK4mE">strongly continuous</a> one-parameter unitary group. Then there exists a unique (possibly unbounded) operator 
  
    
      
        A
        :
        
          
            
              D
            
          
          
            A
          
        
        →
        
          
            H
          
        
      
    
    {\displaystyle A:{\mathcal {D}}_{A}\to {\mathcal {H}}}
  
, that is self-adjoint on 
  
    
      
        
          
            
              D
            
          
          
            A
          
        
      
    
    {\displaystyle {\mathcal {D}}_{A}}
  
 and such that

∀
        t
        ∈
        
          R
        
        :
        
        
          U
          
            t
          
        
        =
        
          e
          
            i
            t
            A
          
        
        .
      
    
    {\displaystyle \forall t\in \mathbb {R} :\qquad U_{t}=e^{itA}.}

The domain of 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is defined by

D
            
          
          
            A
          
        
        =
        
          {
          
            ψ
            ∈
            
              
                H
              
            
            
              |
              
                
                  lim
                  
                    ε
                    →
                    0
                  
                
                
                  
                    
                      −
                      i
                    
                    ε
                  
                
                
                  (
                  
                    
                      U
                      
                        ε
                      
                    
                    (
                    ψ
                    )
                    −
                    ψ
                  
                  )
                
                
                   exists
                
              
              
            
          
          }
        
        .
      
    
    {\displaystyle {\mathcal {D}}_{A}=\left\{\psi \in {\mathcal {H}}\left|\lim _{\varepsilon \to 0}{\frac {-i}{\varepsilon }}\left(U_{\varepsilon }(\psi )-\psi \right){\text{ exists}}\right.\right\}.}

Conversely, let 
  
    
      
        A
        :
        
          
            
              D
            
          
          
            A
          
        
        →
        
          
            H
          
        
      
    
    {\displaystyle A:{\mathcal {D}}_{A}\to {\mathcal {H}}}
  
 be a (possibly unbounded) self-adjoint operator on 
  
    
      
        
          
            
              D
            
          
          
            A
          
        
        ⊆
        
          
            H
          
        
        .
      
    
    {\displaystyle {\mathcal {D}}_{A}\subseteq {\mathcal {H}}.}
  
 Then the one-parameter family 
  
    
      
        (
        
          U
          
            t
          
        
        
          )
          
            t
            ∈
            
              R
            
          
        
      
    
    {\displaystyle (U_{t})_{t\in \mathbb {R} }}
  
 of unitary operators defined by

∀
        t
        ∈
        
          R
        
        :
        
        
          U
          
            t
          
        
        :=
        
          e
          
            i
            t
            A
          
        
      
    
    {\displaystyle \forall t\in \mathbb {R} :\qquad U_{t}:=e^{itA}}

is a strongly continuous one-parameter group.
<p>In both parts of the theorem, the expression 
  
    
      
        
          e
          
            i
            t
            A
          
        
      
    
    {\displaystyle e^{itA}}
  
 is defined by means of the <a href="/facts/Functional_calculus/Y84ugVzS">functional calculus</a>, which uses the <a href="/facts/Spectral_theorem/Vqc04B21">spectral theorem</a> for unbounded <a href="/facts/Self-adjoint_operator/VrH1nEAK">self-adjoint operators</a>. 
</p><p>The operator 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is called the infinitesimal generator of 
  
    
      
        (
        
          U
          
            t
          
        
        
          )
          
            t
            ∈
            
              R
            
          
        
        .
      
    
    {\displaystyle (U_{t})_{t\in \mathbb {R} }.}
  
 Furthermore, 
  
    
      
        A
      
    
    {\displaystyle A}
  
 will be a bounded operator if and only if the operator-valued mapping 
  
    
      
        t
        ↦
        
          U
          
            t
          
        
      
    
    {\displaystyle t\mapsto U_{t}}
  
 is <a href="/facts/Norm_(mathematics)/xIbR4uE1">norm</a>-continuous.
</p><p>The infinitesimal generator 
  
    
      
        A
      
    
    {\displaystyle A}
  
 of a strongly continuous unitary group 
  
    
      
        (
        
          U
          
            t
          
        
        
          )
          
            t
            ∈
            
              R
            
          
        
      
    
    {\displaystyle (U_{t})_{t\in \mathbb {R} }}
  
 may be computed as
</p>

A
        ψ
        =
        −
        i
        
          lim
          
            ε
            →
            0
          
        
        
          
            
              
                U
                
                  ε
                
              
              ψ
              −
              ψ
            
            ε
          
        
        ,
      
    
    {\displaystyle A\psi =-i\lim _{\varepsilon \to 0}{\frac {U_{\varepsilon }\psi -\psi }{\varepsilon }},}

<p>with the domain of 
  
    
      
        A
      
    
    {\displaystyle A}
  
 consisting of those vectors 
  
    
      
        ψ
      
    
    {\displaystyle \psi }
  
 for which the limit exists in the norm topology. That is to say, 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is equal to 
  
    
      
        −
        i
      
    
    {\displaystyle -i}
  
 times the derivative of 
  
    
      
        
          U
          
            t
          
        
      
    
    {\displaystyle U_{t}}
  
 with respect to 
  
    
      
        t
      
    
    {\displaystyle t}
  
 at 
  
    
      
        t
        =
        0
      
    
    {\displaystyle t=0}
  
. Part of the statement of the theorem is that this derivative exists—i.e., that 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is a densely defined self-adjoint operator. The result is not obvious even in the finite-dimensional case, since 
  
    
      
        
          U
          
            t
          
        
      
    
    {\displaystyle U_{t}}
  
 is only assumed (ahead of time) to be continuous, and not differentiable.
</p>
<h2 id="example">Example</h2>
<p>The family of translation operators
</p>

[
          
            
              T
              
                t
              
            
            (
            ψ
            )
          
          ]
        
        (
        x
        )
        =
        ψ
        (
        x
        +
        t
        )
      
    
    {\displaystyle \left[T_{t}(\psi )\right](x)=\psi (x+t)}

<p>is a one-parameter unitary group of unitary operators; the infinitesimal generator of this family is an <a href="/facts/Extensions_of_symmetric_operators/HN6RGnde">extension</a> of the differential operator
</p>

−
        i
        
          
            d
            
              d
              x
            
          
        
      
    
    {\displaystyle -i{\frac {d}{dx}}}

<p>defined on the space of continuously differentiable complex-valued functions with <a href="/facts/Compact_support/BTuMGbsb">compact support</a> on 
  
    
      
        
          R
        
        .
      
    
    {\displaystyle \mathbb {R} .}
  
 Thus
</p>

T
          
            t
          
        
        =
        
          e
          
            t
            
              
                d
                
                  d
                  x
                
              
            
          
        
        .
      
    
    {\displaystyle T_{t}=e^{t{\frac {d}{dx}}}.}

<p>In other words, motion on the line is generated by the <a href="/facts/Momentum_operator/nPx4UL3k">momentum operator</a>.
</p>
<h2 id="applications">Applications</h2>
<p>Stone's theorem has numerous applications in <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>. For instance, given an isolated quantum mechanical system, with Hilbert space of states H, <a href="/facts/Time_evolution/8XxTc0fN">time evolution</a> is a strongly continuous one-parameter unitary group on 
  
    
      
        
          
            H
          
        
      
    
    {\displaystyle {\mathcal {H}}}
  
. The infinitesimal generator of this group is the system <a href="/facts/Hamiltonian_(quantum_theory)/5XwK2HmD">Hamiltonian</a>.
</p>
<p class="note">Further information: <a href="/facts/Stone%25E2%2580%2593von_Neumann_theorem/4C64mKYR">Stone–von Neumann theorem</a> and <a href="/facts/Heisenberg_group/rsqq1426">Heisenberg group</a></p>
<h2 id="using-fourier-transform">Using Fourier transform</h2>
<p>Stone's Theorem can be recast using the language of the <a href="/facts/Fourier_transform/XXhKJtc8">Fourier transform</a>. The real line 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
 is a locally compact abelian group. Non-degenerate *-representations of the <a href="/facts/Group_algebra_of_a_locally_compact_group/I1BVHqYD">group C*-algebra</a> 
  
    
      
        
          C
          
            ∗
          
        
        (
        
          R
        
        )
      
    
    {\displaystyle C^{*}(\mathbb {R} )}
  
 are in one-to-one correspondence with strongly continuous unitary representations of 
  
    
      
        
          R
        
        ,
      
    
    {\displaystyle \mathbb {R} ,}
  
 i.e., strongly continuous one-parameter unitary groups. On the other hand, the Fourier transform is a *-isomorphism from 
  
    
      
        
          C
          
            ∗
          
        
        (
        
          R
        
        )
      
    
    {\displaystyle C^{*}(\mathbb {R} )}
  
 to 
  
    
      
        
          C
          
            0
          
        
        (
        
          R
        
        )
        ,
      
    
    {\displaystyle C_{0}(\mathbb {R} ),}
  
 the 
  
    
      
        
          C
          
            ∗
          
        
      
    
    {\displaystyle C^{*}}
  
-algebra of continuous complex-valued functions on the real line that vanish at infinity. Hence, there is a one-to-one correspondence between strongly continuous one-parameter unitary groups and *-representations of 
  
    
      
        
          C
          
            0
          
        
        (
        
          R
        
        )
        .
      
    
    {\displaystyle C_{0}(\mathbb {R} ).}
  
 As every *-representation of 
  
    
      
        
          C
          
            0
          
        
        (
        
          R
        
        )
      
    
    {\displaystyle C_{0}(\mathbb {R} )}
  
 corresponds uniquely to a self-adjoint operator, Stone's Theorem holds.
</p><p>Therefore, the procedure for obtaining the infinitesimal generator of a strongly continuous one-parameter unitary group is as follows:
</p>
<ul><li>Let 
  
    
      
        (
        
          U
          
            t
          
        
        
          )
          
            t
            ∈
            
              R
            
          
        
      
    
    {\displaystyle (U_{t})_{t\in \mathbb {R} }}
  
 be a strongly continuous unitary representation of 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
 on a <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> 
  
    
      
        
          
            H
          
        
      
    
    {\displaystyle {\mathcal {H}}}
  
.</li>
<li>Integrate this unitary representation to yield a non-degenerate *-representation 
  
    
      
        ρ
      
    
    {\displaystyle \rho }
  
 of 
  
    
      
        
          C
          
            ∗
          
        
        (
        
          R
        
        )
      
    
    {\displaystyle C^{*}(\mathbb {R} )}
  
 on 
  
    
      
        
          
            H
          
        
      
    
    {\displaystyle {\mathcal {H}}}
  
 by first defining 
  
    
      
        ∀
        f
        ∈
        
          C
          
            c
          
        
        (
        
          R
        
        )
        :
        
        ρ
        (
        f
        )
        :=
        
          ∫
          
            
              R
            
          
        
        f
        (
        t
        )
         
        
          U
          
            t
          
        
        d
        t
        ,
      
    
    {\displaystyle \forall f\in C_{c}(\mathbb {R} ):\quad \rho (f):=\int _{\mathbb {R} }f(t)~U_{t}dt,}
  
 and then extending 
  
    
      
        ρ
      
    
    {\displaystyle \rho }
  
 to all of 
  
    
      
        
          C
          
            ∗
          
        
        (
        
          R
        
        )
      
    
    {\displaystyle C^{*}(\mathbb {R} )}
  
 by continuity.</li>
<li>Use the Fourier transform to obtain a non-degenerate *-representation 
  
    
      
        τ
      
    
    {\displaystyle \tau }
  
 of 
  
    
      
        
          C
          
            0
          
        
        (
        
          R
        
        )
      
    
    {\displaystyle C_{0}(\mathbb {R} )}
  
 on 
  
    
      
        
          
            H
          
        
      
    
    {\displaystyle {\mathcal {H}}}
  
.</li>
<li>By the <a href="/facts/Riesz%25E2%2580%2593Markov%25E2%2580%2593Kakutani_representation_theorem/vtJ7grhH">Riesz-Markov Theorem</a>, 
  
    
      
        τ
      
    
    {\displaystyle \tau }
  
 gives rise to a <a href="/facts/Projection-valued_measure/baNsfTzA">projection-valued measure</a> on 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
 that is the resolution of the identity of a unique <a href="/facts/Self-adjoint_operator/VrH1nEAK">self-adjoint operator</a> 
  
    
      
        A
      
    
    {\displaystyle A}
  
, which may be unbounded.</li>
<li>Then 
  
    
      
        A
      
    
    {\displaystyle A}
  
 is the infinitesimal generator of 
  
    
      
        (
        
          U
          
            t
          
        
        
          )
          
            t
            ∈
            
              R
            
          
        
        .
      
    
    {\displaystyle (U_{t})_{t\in \mathbb {R} }.}
  
</li></ul>
<p>The precise definition of 
  
    
      
        
          C
          
            ∗
          
        
        (
        
          R
        
        )
      
    
    {\displaystyle C^{*}(\mathbb {R} )}
  
 is as follows. Consider the *-algebra 
  
    
      
        
          C
          
            c
          
        
        (
        
          R
        
        )
        ,
      
    
    {\displaystyle C_{c}(\mathbb {R} ),}
  
 the continuous complex-valued functions on 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
 with compact support, where the multiplication is given by <a href="/facts/Convolution/PVPrdz9J">convolution</a>. The completion of this *-algebra with respect to the 
  
    
      
        
          L
          
            1
          
        
      
    
    {\displaystyle L^{1}}
  
-norm is a Banach *-algebra, denoted by 
  
    
      
        (
        
          L
          
            1
          
        
        (
        
          R
        
        )
        ,
        ⋆
        )
        .
      
    
    {\displaystyle (L^{1}(\mathbb {R} ),\star ).}
  
 Then 
  
    
      
        
          C
          
            ∗
          
        
        (
        
          R
        
        )
      
    
    {\displaystyle C^{*}(\mathbb {R} )}
  
 is defined to be the enveloping 
  
    
      
        
          C
          
            ∗
          
        
      
    
    {\displaystyle C^{*}}
  
-algebra of 
  
    
      
        (
        
          L
          
            1
          
        
        (
        
          R
        
        )
        ,
        ⋆
        )
      
    
    {\displaystyle (L^{1}(\mathbb {R} ),\star )}
  
, i.e., its completion with respect to the largest possible 
  
    
      
        
          C
          
            ∗
          
        
      
    
    {\displaystyle C^{*}}
  
-norm. It is a non-trivial fact that, via the Fourier transform, 
  
    
      
        
          C
          
            ∗
          
        
        (
        
          R
        
        )
      
    
    {\displaystyle C^{*}(\mathbb {R} )}
  
 is isomorphic to 
  
    
      
        
          C
          
            0
          
        
        (
        
          R
        
        )
        .
      
    
    {\displaystyle C_{0}(\mathbb {R} ).}
  
 A result in this direction is the <a href="/facts/Riemann%25E2%2580%2593Lebesgue_lemma/Dd03bg0z">Riemann-Lebesgue Lemma</a>, which says that the Fourier transform maps 
  
    
      
        
          L
          
            1
          
        
        (
        
          R
        
        )
      
    
    {\displaystyle L^{1}(\mathbb {R} )}
  
 to 
  
    
      
        
          C
          
            0
          
        
        (
        
          R
        
        )
        .
      
    
    {\displaystyle C_{0}(\mathbb {R} ).}

</p>
<h2 id="generalizations">Generalizations</h2>
<p>The <a href="/facts/Stone%25E2%2580%2593von_Neumann_theorem/4C64mKYR">Stone–von Neumann theorem</a> generalizes Stone's theorem to a <i>pair</i> of self-adjoint operators, 
  
    
      
        (
        P
        ,
        Q
        )
      
    
    {\displaystyle (P,Q)}
  
, satisfying the <a href="/facts/Canonical_commutation_relation/UgUW6TuT">canonical commutation relation</a>, and shows that these are all unitarily equivalent to the <a href="/facts/Position_operator/zQUM4fIk">position operator</a> and <a href="/facts/Momentum_operator/nPx4UL3k">momentum operator</a> on 
  
    
      
        
          L
          
            2
          
        
        (
        
          R
        
        )
        .
      
    
    {\displaystyle L^{2}(\mathbb {R} ).}

</p><p>The <a href="/facts/Hille%25E2%2580%2593Yosida_theorem/s1qSoQuq">Hille–Yosida theorem</a> generalizes Stone's theorem to strongly continuous one-parameter semigroups of <a href="/facts/Contraction_(operator_theory)/W4UcMlRd">contractions</a> on <a href="/facts/Banach_space/sDEIk7EC">Banach spaces</a>.
</p>

<h2 id="bibliography">Bibliography</h2>
<ul><li>Hall, B.C. (2013), <i>Quantum Theory for Mathematicians</i>, 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>
<li>von Neumann, John (1932), "Über einen Satz von Herrn M. H. Stone", <i><a href="/facts/Annals_of_Mathematics/CUYktUqD">Annals of Mathematics</a></i>, Second Series (in German), 33 (3), Annals of Mathematics: 567–573, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1968535">10.2307/1968535</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0003-486X">0003-486X</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1968535">1968535</a></li>
<li>Stone, M. H. (1930), "Linear Transformations in Hilbert Space. III. Operational Methods and Group Theory", <i><a href="/facts/Proceedings_of_the_National_Academy_of_Sciences/w75CThX0">Proceedings of the National Academy of Sciences of the United States of America</a></i>, 16 (2), National Academy of Sciences: 172–175, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1930PNAS...16..172S">1930PNAS...16..172S</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1073%2Fpnas.16.2.172">10.1073/pnas.16.2.172</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0027-8424">0027-8424</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/85485">85485</a>, <a href="/facts/PMC_(identifier)/dX1zMt71">PMC</a> <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1075964">1075964</a>, <a href="/facts/PMID_(identifier)/JlHAvMHt">PMID</a> <a href="https://pubmed.ncbi.nlm.nih.gov/16587545">16587545</a></li>
<li>Stone, M. H. (1932), "On one-parameter unitary groups in Hilbert Space", <i>Annals of Mathematics</i>, 33 (3): 643–648, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F1968538">10.2307/1968538</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/1968538">1968538</a></li>
<li>K. Yosida, <i>Functional Analysis</i>, Springer-Verlag, (1968)</li></ul>

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

<ol>
<li id="fn:1"><p>Hall 2013 Theorem 10.15 - Hall, B.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:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Stone's theorem on one-parameter unitary groups open-in-new

Stone's theorem on one-parameter unitary groups