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CCR and CAR algebras
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<h2 id="ccr-and-car-as--algebras">CCR and CAR as *-algebras</h2> <p>Let V {\displaystyle V} be a <a href="/facts/Real_number/R02gw5Pb">real</a> <a href="/facts/Vector_space/M1pxMLx2">vector space</a> equipped with a <a href="/facts/Algebraic_curve/LnWsPmDP">nonsingular</a> real <a href="/facts/Skew-symmetric_matrix/nw5CQeLN">antisymmetric</a> <a href="/facts/Bilinear_form/gyvzn9ym">bilinear form</a> ( ⋅ , ⋅ ) {\displaystyle (\cdot ,\cdot )} (i.e. a <a href="/facts/Symplectic_vector_space/o95tWKBc">symplectic vector space</a>). The <a href="/facts/Unital_algebra/8T8ulWJb">unital</a> <a href="/facts/*-algebra/vcE5TWnT">*-algebra</a> generated by elements of V {\displaystyle V} subject to the relations </p> f g − g f = i ( f , g ) {\displaystyle fg-gf=i(f,g)\,} f ∗ = f , {\displaystyle f^{*}=f,\,} <p>for any f , g {\displaystyle f,~g} in V {\displaystyle V} is called the canonical commutation relations (CCR) algebra. The uniqueness of the representations of this algebra when V {\displaystyle V} is <a href="/facts/Finite_dimensional/z8m02A3W">finite dimensional</a> is discussed in the <a href="/facts/Stone%E2%80%93von_Neumann_theorem/4C64mKYR">Stone–von Neumann theorem</a>. </p><p>If V {\displaystyle V} is equipped with a <a href="/facts/Algebraic_curve/LnWsPmDP">nonsingular</a> real <a href="/facts/Symmetric_bilinear_form/UI9b6RK7">symmetric bilinear form</a> ( ⋅ , ⋅ ) {\displaystyle (\cdot ,\cdot )} instead, the unital *-algebra generated by the elements of V {\displaystyle V} subject to the relations </p> f g + g f = ( f , g ) , {\displaystyle fg+gf=(f,g),\,} f ∗ = f , {\displaystyle f^{*}=f,\,} <p>for any f , g {\displaystyle f,~g} in V {\displaystyle V} is called the canonical anticommutation relations (CAR) algebra. </p> <h2 id="the-c-algebra-of-ccr">The C*-algebra of CCR</h2> <p>There is a distinct, but closely related meaning of CCR algebra, called the CCR C*-algebra. Let H {\displaystyle H} be a real symplectic vector space with nonsingular symplectic form ( ⋅ , ⋅ ) {\displaystyle (\cdot ,\cdot )} . In the theory of <a href="/facts/Operator_algebra/4fycDNQ1">operator algebras</a>, the CCR algebra over H {\displaystyle H} is the unital <a href="/facts/C*-algebra/B3tnGHIo">C*-algebra</a> generated by elements { W ( f ) : f ∈ H } {\displaystyle \{W(f):~f\in H\}} subject to </p> W ( f ) W ( g ) = e − i ( f , g ) W ( f + g ) , {\displaystyle W(f)W(g)=e^{-i(f,g)}W(f+g),\,} W ( f ) ∗ = W ( − f ) . {\displaystyle W(f)^{*}=W(-f).\,} <p>These are called the Weyl form of the canonical commutation relations and, in particular, they imply that each W ( f ) {\displaystyle W(f)} is <a href="/facts/Unitary_operator/rkypfQQ9">unitary</a> and W ( 0 ) = 1 {\displaystyle W(0)=1} . It is well known that the CCR algebra is a simple (unless the sympletic form is degenerate) non-separable algebra and is unique up to isomorphism.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p><p>When H {\displaystyle H} is a complex <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> and ( ⋅ , ⋅ ) {\displaystyle (\cdot ,\cdot )} is given by the imaginary part of the inner-product, the CCR algebra is <a href="/facts/Representation_theory/3ydLtaGH">faithfully represented</a> on the <a href="/facts/Fock_space/I2PYLlhl">symmetric Fock space</a> over H {\displaystyle H} by setting </p> W ( f ) ( 1 , g , g ⊗ 2 2 ! , g ⊗ 3 3 ! , … ) = e − 1 2 ‖ f ‖ 2 − ⟨ f , g ⟩ ( 1 , f + g , ( f + g ) ⊗ 2 2 ! , ( f + g ) ⊗ 3 3 ! , … ) , {\displaystyle W(f)\left(1,g,{\frac {g^{\otimes 2}}{2!}},{\frac {g^{\otimes 3}}{3!}},\ldots \right)=e^{-{\frac {1}{2}}\|f\|^{2}-\langle f,g\rangle }\left(1,f+g,{\frac {(f+g)^{\otimes 2}}{2!}},{\frac {(f+g)^{\otimes 3}}{3!}},\ldots \right),} <p>for any f , g ∈ H {\displaystyle f,g\in H} . The field operators B ( f ) {\displaystyle B(f)} are defined for each f ∈ H {\displaystyle f\in H} as the <a href="/facts/Stone%27s_theorem_on_one-parameter_unitary_groups/1quJvzU7">generator</a> of the one-parameter unitary group ( W ( t f ) ) t ∈ R {\displaystyle (W(tf))_{t\in \mathbb {R} }} on the symmetric Fock space. These are <a href="/facts/Self-adjoint_operator/VrH1nEAK">self-adjoint</a> <a href="/facts/Unbounded_operator/5u2njqHt">unbounded operators</a>, however they formally satisfy </p> B ( f ) B ( g ) − B ( g ) B ( f ) = 2 i Im ⟨ f , g ⟩ . {\displaystyle B(f)B(g)-B(g)B(f)=2i\operatorname {Im} \langle f,g\rangle .} <p>As the assignment f ↦ B ( f ) {\displaystyle f\mapsto B(f)} is real-linear, so the operators B ( f ) {\displaystyle B(f)} define a CCR algebra over ( H , 2 Im ⟨ ⋅ , ⋅ ⟩ ) {\displaystyle (H,2\operatorname {Im} \langle \cdot ,\cdot \rangle )} in the sense of Section 1. </p> <h2 id="the-c-algebra-of-car">The C*-algebra of CAR</h2> <p>Let H {\displaystyle H} be a Hilbert space. In the theory of operator algebras the CAR algebra is the unique <a href="/facts/Complete_metric_space/lsckmU8G">C*-completion</a> of the complex unital *-algebra generated by elements { b ( f ) , b ∗ ( f ) : f ∈ H } {\displaystyle \{b(f),b^{*}(f):~f\in H\}} subject to the relations </p> b ( f ) b ∗ ( g ) + b ∗ ( g ) b ( f ) = ⟨ f , g ⟩ , {\displaystyle b(f)b^{*}(g)+b^{*}(g)b(f)=\langle f,g\rangle ,\,} b ( f ) b ( g ) + b ( g ) b ( f ) = 0 , {\displaystyle b(f)b(g)+b(g)b(f)=0,\,} λ b ∗ ( f ) = b ∗ ( λ f ) , {\displaystyle \lambda b^{*}(f)=b^{*}(\lambda f),\,} b ( f ) ∗ = b ∗ ( f ) , {\displaystyle b(f)^{*}=b^{*}(f),\,} <p>for any f , g ∈ H {\displaystyle f,g\in H} , λ ∈ C {\displaystyle \lambda \in \mathbb {C} } . When H {\displaystyle H} is separable the CAR algebra is an <a href="/facts/Approximately_finite_dimensional_C*-algebra/9QUzbEwk">AF algebra</a> and in the special case H {\displaystyle H} is infinite dimensional it is often written as M 2 ∞ ( C ) {\displaystyle {M_{2^{\infty }}(\mathbb {C} )}} .<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p><p>Let F a ( H ) {\displaystyle F_{a}(H)} be the <a href="/facts/Fock_space/I2PYLlhl">antisymmetric Fock space</a> over H {\displaystyle H} and let P a {\displaystyle P_{a}} be the orthogonal projection onto antisymmetric vectors: </p> P a : ⨁ n = 0 ∞ H ⊗ n → F a ( H ) . {\displaystyle P_{a}:\bigoplus _{n=0}^{\infty }H^{\otimes n}\to F_{a}(H).\,} <p>The CAR algebra is faithfully represented on F a ( H ) {\displaystyle F_{a}(H)} by setting </p> b ∗ ( f ) P a ( g 1 ⊗ g 2 ⊗ ⋯ ⊗ g n ) = P a ( f ⊗ g 1 ⊗ g 2 ⊗ ⋯ ⊗ g n ) {\displaystyle b^{*}(f)P_{a}(g_{1}\otimes g_{2}\otimes \cdots \otimes g_{n})=P_{a}(f\otimes g_{1}\otimes g_{2}\otimes \cdots \otimes g_{n})\,} <p>for all f , g 1 , … , g n ∈ H {\displaystyle f,g_{1},\ldots ,g_{n}\in H} and n ∈ N {\displaystyle n\in \mathbb {N} } . The fact that these form a C*-algebra is due to the fact that creation and annihilation operators on antisymmetric Fock space are bona-fide <a href="/facts/Bounded_operator/LYmDTIqR">bounded operators</a>. Moreover, the field operators B ( f ) := b ∗ ( f ) + b ( f ) {\displaystyle B(f):=b^{*}(f)+b(f)} satisfy </p> B ( f ) B ( g ) + B ( g ) B ( f ) = 2 R e ⟨ f , g ⟩ , {\displaystyle B(f)B(g)+B(g)B(f)=2\mathrm {Re} \langle f,g\rangle ,\,} <p>giving the relationship with Section 1. </p> <h2 id="superalgebra-generalization">Superalgebra generalization</h2> <p>Let V {\displaystyle V} be a real Z 2 {\displaystyle \mathbb {Z} _{2}} -<a href="/facts/Graded_vector_space/Ff1maywV">graded vector space</a> equipped with a nonsingular antisymmetric bilinear superform ( ⋅ , ⋅ ) {\displaystyle (\cdot ,\cdot )} (i.e. ( g , f ) = − ( − 1 ) | f | | g | ( f , g ) {\displaystyle (g,f)=-(-1)^{|f||g|}(f,g)} ) such that ( f , g ) {\displaystyle (f,g)} is real if either f {\displaystyle f} or g {\displaystyle g} is an even element and <a href="/facts/Imaginary_number/6Xh9W3hq">imaginary</a> if both of them are odd. The unital *-algebra generated by the elements of V {\displaystyle V} subject to the relations </p> f g − ( − 1 ) | f | | g | g f = i ( f , g ) {\displaystyle fg-(-1)^{|f||g|}gf=i(f,g)\,} f ∗ = f , g ∗ = g {\displaystyle f^{*}=f,~g^{*}=g\,} <p>for any two pure elements f , g {\displaystyle f,~g} in V {\displaystyle V} is the obvious <a href="/facts/Superalgebra/96vSHP12">superalgebra</a> generalization which unifies CCRs with CARs: if all pure elements are even, one obtains a CCR, while if all pure elements are odd, one obtains a CAR. </p><p>In mathematics, the abstract structure of the CCR and CAR algebras, over any field, not just the complex numbers, is studied by the name of <a href="/facts/Weyl_algebra/PADSD0k9">Weyl</a> and <a href="/facts/Clifford_algebra/qnqSCz5h">Clifford algebras</a>, where many significant results have accrued. One of these is that the <a href="/facts/Graded_algebra/AIUjkSaP">graded</a> generalizations of <a href="/facts/Weyl_algebra/PADSD0k9">Weyl</a> and <a href="/facts/Clifford_algebra/qnqSCz5h">Clifford</a> algebras allow the basis-free formulation of the canonical commutation and anticommutation relations in terms of a symplectic and a symmetric non-degenerate bilinear form. In addition, the binary elements in this graded Weyl algebra give a basis-free version of the commutation relations of the <a href="/facts/Symplectic_Lie_algebra/G4CKBc7z">symplectic</a> and <a href="/facts/Indefinite_orthogonal_group/snKhwDB2">indefinite orthogonal Lie algebras</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Bose%E2%80%93Einstein_statistics/Utb2jN3n">Bose–Einstein statistics</a></li> <li><a href="/facts/Fermi%E2%80%93Dirac_statistics/aLo4bton">Fermi–Dirac statistics</a></li> <li><a href="/facts/Glossary_of_string_theory/s0rfBHQ9">Glossary of string theory</a></li> <li><a href="/facts/Heisenberg_group/rsqq1426">Heisenberg group</a></li> <li><a href="/facts/Bogoliubov_transformation/TxMcRVyG">Bogoliubov transformation</a></li> <li><a href="/facts/(%E2%88%921)F/S61nJqKb">(−1)F</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Bratteli, Ola; Robinson, Derek W. (1997). Operator Algebras and Quantum Statistical Mechanics: v.2. Springer, 2nd ed. ISBN 978-3-540-61443-2. <a href="978-3-540-61443-2" target="_blank">978-3-540-61443-2</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Petz, Denes (1990). An Invitation to the Algebra of Canonical Commutation Relations. Leuven University Press. ISBN 978-90-6186-360-1. <a href="978-90-6186-360-1" target="_blank">978-90-6186-360-1</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Evans, David E.; Kawahigashi, Yasuyuki (1998). Quantum Symmetries in Operator Algebras. Oxford University Press. ISBN 978-0-19-851175-5.. <a href="978-0-19-851175-5" target="_blank">978-0-19-851175-5</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Roger Howe (1989). "Remarks on Classical Invariant Theory". Transactions of the American Mathematical Society. 313 (2): 539–570. doi:10.1090/S0002-9947-1989-0986027-X. JSTOR 2001418. <a href="https://doi.org/10.1090%2FS0002-9947-1989-0986027-X" target="_blank">https://doi.org/10.1090%2FS0002-9947-1989-0986027-X</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>