Bargman - Wigner Theorem

The Bargman - Wigner theorem is a theorem of axiomatic quantum field theory. It reveals the meaning of the concept of a universal covering group under Poincare transformations in relativistic quantum theory. It was proved by Yu. Wigner ^[1] and V. Bargman ^[2] .

Content

Wording

The state vectors during transformations from the Poincare’s own group are transformed according to the unitary representation of its universal covering (the quantum-mechanical Poincare’s own group) ^[3] .

In other words, from each ray $T(a,\Lambda )$ ${\ displaystyle T (a, \ Lambda)}$ you can select one representative $U(a,\Lambda )\in T(a,\Lambda )$ ${\ displaystyle U (a, \ Lambda) \ in T (a, \ Lambda)}$ so that there are relations ^[4] :

$U(0,1)=1,U({\underline {a_{1}}},A_{1})U({\underline {a_{2}}},A_{2})=U({\underline {a_{1}}}+A_{1}^{*}a_{2}A_{1},A_{1}A_{2})$ ${\ displaystyle U (0,1) = 1, U ({\ underline {a_ {1}}}, A_ {1}) U ({\ underline {a_ {2}}}, A_ {2}) = U ({\ underline {a_ {1}}} + A_ {1} ^ {*} a_ {2} A_ {1}, A_ {1} A_ {2})}$

Where $a$ ${\ displaystyle a}$ defined by the formula $x=x^{\alpha }{\sigma }_{\alpha }=x^{0}\sigma _{0}+x^{1}\sigma _{1}+x^{2}\sigma _{2}+x^{3}\sigma _{3}={\begin{pmatrix}x^{0}+x^{3}&x^{1}-ix^{2}\\x^{1}+ix^{2}&x^{0}-x^{3}\end{pmatrix}}$ ${\ displaystyle x = x ^ {\ alpha} {\ sigma} _ {\ alpha} = x ^ {0} \ sigma _ {0} + x ^ {1} \ sigma _ {1} + x ^ {2} \ sigma _ {2} + x ^ {3} \ sigma _ {3} = {\ begin {pmatrix} x ^ {0} + x ^ {3} & x ^ {1} -ix ^ {2} \\ x ^ {1} + ix ^ {2} & x ^ {0} -x ^ {3} \ end {pmatrix}}}$ .

Explanation

A ray is a state vector in a separable Hilbert space ^[5] . Group $G_{2}$ ${\ displaystyle G_ {2}}$ called the universal covering connected group $G_{1}$ ${\ displaystyle G_ {1}}$ , if a $G_{2}$ ${\ displaystyle G_ {2}}$ is a minimal simply connected group homomorphic $G_{1}$ ${\ displaystyle G_ {1}}$ ^[6] . $x$ ${\ displaystyle x}$ is a four-dimensional vector ^[7] . $\sigma _{0},...,\sigma _{3}$ ${\ displaystyle \ sigma _ {0}, ..., \ sigma _ {3}}$ - Pauli matrices ^[7] .

Notes

↑ Wigner EP On unitary representations of the inhomogenous Lorentz group // Annals of Mathematics . - 1939. - T. 40. - PP. 150-204. - URL: https://www.jstor.org/stable/1968551
↑ Bargmann V. On Unitary Ray Representations of Continuous Groups // Annals of Mathematics . - 1954. - T. 59. - S. 1–46. - URL: https://www.jstor.org/stable/1969831
↑ Bogolyubov, 1969 , p. 106.
↑ Bogolyubov, 1969 , p. 105.
↑ Bogolyubov, 1969 , p. 85.
↑ Bogolyubov, 1969 , p. 101.
↑ ¹ ² Bogolyubov, 1969 , p. 99.

Literature

Bogolyubov N.N. , Logunov A.A. , Todorov I.T. Fundamentals of the axiomatic approach in quantum field theory. - M .: Nauka, 1969 .-- 424 p.

[1] Wigner EP On unitary representations of the inhomogenous Lorentz group // Annals of Mathematics . - 1939. - T. 40. - PP. 150-204. - URL: https://www.jstor.org/stable/1968551

[2] Bargmann V. On Unitary Ray Representations of Continuous Groups // Annals of Mathematics . - 1954. - T. 59. - S. 1–46. - URL: https://www.jstor.org/stable/1969831

[_b9f6dd5da3ae1b72-3] Bogolyubov, 1969 , p. 106.

[_b9f6dd5da3ae1b71-4] Bogolyubov, 1969 , p. 105.

[_7ddbb74fa9ddbcd2-5] Bogolyubov, 1969 , p. 85.

[_b9f6dd5da3ae1b75-6] Bogolyubov, 1969 , p. 101.

[_7ddbb64fa9ddbb0d-7] ¹ ² Bogolyubov, 1969 , p. 99.