Generalized Wigner Operators and Relativistic Gauge Fields

Abstract

We introduce and study the generalized Wigner operator. By definition, such an operator transforms the Wigner wave function into a local relativistic field corresponding to an irreducible representation of the Poincaré group by extended discrete transformations, with integer helicities $\lambda$ and $- \lambda$. It is shown that the relativistic fields constructed in this way are gauge potentials and satisfy the relations that determine free massless higher spin fields.

Appendices

APPENDIX 1

1.1 COVERING OF THE $4D$ LORENTZ GROUP. WIGNER OPERATORS

The relationship between the Lorentz group and its covering group $SL(2,\mathbb{C})$ is formulated in the frame work of space $\mathcal{H}$ of Hermitian $(2 \times 2)$ matrices. In this case the basis matrices are ${{\sigma }^{0}} = {{I}_{2}}$ and ${{\sigma }^{i}}$, $i = 1,2,3$ are the $\sigma$-Pauli matrices. These matrices establish a one-to-one correspondence between the set of vectors in the space ${{x}_{\mu }} \in {{\mathbb{R}}^{{1,3}}}$ and the set $\mathcal{H}$ of Hermitian matrices $X = {{x}_{\mu }}{{\sigma }^{\mu }} \in \mathcal{H}$. The action of the group $SL(2,\mathbb{C})$ on the set $\mathcal{H}$

$$X \to X{\kern 1pt} ' = AX{{A}^{\dag }}{\kern 1pt} ,\;\;\;X,X{\kern 1pt} ' \in \mathcal{H}{\kern 1pt} ,\;\;\;A \in SL(2,\mathbb{C})$$

(A.1)

leads to the group homomorphism $SL(2,\mathbb{C}) \to S{{O}^{ \uparrow }}(1,3)$ represented in (2.2).

As a test 4-momentum of a massless particle, we take the vector $\mathop p\limits^\circ {\kern 1pt} \in {{\mathbb{R}}^{{1,3}}}$, which has the following components

$$\left| {\left| {{{{\mathop p\limits^\circ }}_{\mu }}} \right|} \right| = ({{\mathop p\limits^\circ }_{0}},{{\mathop p\limits^\circ }_{1}},{{\mathop p\limits^\circ }_{2}},{{\mathop p\limits^\circ }_{3}}) = (E,0,0,E){\kern 1pt} .$$

(A.2)

The Wigner operator ${{A}_{{(p)}}} \in SL(2,\mathbb{C})$ is defined by the matrix equation

$${{A}_{{(p)}}}(\mathop p\limits^\circ {\kern 1pt} \sigma )A_{{(p)}}^{\dag } = (p{\kern 1pt} \sigma ){\kern 1pt} ,$$

(A.3)

where $(x\sigma ): = {{x}_{\mu }}{{\sigma }^{\mu }}$. The arbitrariness in the definition of Wigner operators is fixed by the equality ${{A}_{{(\mathop p\limits^\circ )}}} = {{I}_{2}}$. Relation (A.3) in vector representation has the form $\Lambda _{\nu }^{\mu }({{A}_{{(p)}}}){\kern 1pt} {{\mathop p\limits^\circ }_{\mu }} = {{p}_{\nu }}$. The matrices $h \in SL(2,\mathbb{C})$ from the stability subgroup ${{G}_{{\mathop p\limits^\circ }}}$ preserve the test momentum,

$$h\;(\mathop p\limits^\circ \sigma )\;{{h}^{\dag }} = (\mathop p\limits^\circ {\kern 1pt} \sigma ){\kern 1pt} .$$

(A.4)

In the case of an isotropic 4-momentum (A.2), the elements $h \in {{G}_{{\mathop p\limits^\circ }}} \simeq ISO(2)$ have the form

$$h = \left( {\begin{array}{*{20}{c}} 1&{{{b}_{1}} + i{{b}_{2}}} \\ 0&1 \end{array}} \right){\kern 1pt} \left( {\begin{array}{*{20}{c}} {{{e}^{{\frac{i}{2}\theta }}}}&0 \\ 0&{{{e}^{{ - \frac{i}{2}\theta }}}} \end{array}} \right){\kern 1pt} ;$$

(A.5)

i.e., the matrix $h$ depends on three parameters $\theta \in [0,2\pi ]$ and $\vec {b} = ({{b}_{1}},{{b}_{2}}) \in {{\mathbb{R}}^{2}}$.

The Wigner operator ${{A}_{{(p)}}}$ is defined up to right multiplication by an element from the stability subgroup ${{G}_{{\mathop p\limits^\circ }}}$ and thus parameterizes the coset space $SL(2,\mathbb{C})/{{G}_{{\mathop p\limits^\circ }}}$. The relation $A{\kern 1pt} {{A}_{{(p)}}} = {{A}_{{(\Lambda p)}}}{\kern 1pt} {{h}_{{A,p}}}$ defining the action of the element $A \in SL(2,\mathbb{C})$ on the coset space $SL(2,\mathbb{C})/{{G}_{{\mathop p\limits^\circ }}}$ parameterized by the Wigner operator leads to two consequences

$${{h}_{{A,p}}} = A_{{(\Lambda p)}}^{{ - 1}}{\kern 1pt} A{\kern 1pt} {{A}_{{(p)}}}\;\;\;\; \Rightarrow \;\;\;\;{{h}_{{A,{{\Lambda }^{{ - 1}}}p}}} = A_{{(p)}}^{{ - 1}}{\kern 1pt} A{\kern 1pt} {{A}_{{({{\Lambda }^{{ - 1}}}p)}}}{\kern 1pt} ,$$

(A.6)

where the indices at ${{h}_{{A,p}}}$ indicate that the element of the test momentum stability subgroup depends on $A \in SL(2,\mathbb{C})$ and the 4-momentum $p$.

The parameters that correspond to the stability subgroup element ${{h}_{{A,p}}}$ specified in (A.6) are denoted by ${{\theta }_{{A,p}}}$ and ${{\vec {b}}_{{A,p}}}$ and are determined from the relation

$${{h}_{{A,p}}} = A_{{(\Lambda p)}}^{{ - 1}}{\kern 1pt} A{\kern 1pt} {{A}_{{(p)}}} = \left( {\begin{array}{*{20}{c}} 1&{{{{\mathbf{b}}}_{{A,p}}}} \\ 0&1 \end{array}} \right){\kern 1pt} \left( {\begin{array}{*{20}{c}} {{{e}^{{\frac{i}{2}{{\theta }_{{A,p}}}}}}}&0 \\ 0&{{{e}^{{ - \frac{i}{2}{{\theta }_{{A,p}}}}}}} \end{array}} \right){\kern 1pt} ,$$

(A.7)

where ${{{\mathbf{b}}}_{{A,p}}} = ({{b}_{{A,p}}}{{)}_{1}} + i{{({{b}_{{A,p}}})}_{2}}$. The parameters ${{\theta }_{{A,{{\Lambda }^{{ - 1}}}p}}}$ and ${{\vec {b}}_{{A,{{\Lambda }^{{ - 1}}}p}}}$ corresponding to the second relation from (A.6), are determined from ${{\theta }_{{A,p}}}$ and ${{\vec {b}}_{{A,p}}}$ by the substitution: $p \to \Lambda {{(A)}^{{ - 1}}}p$. Taking into account (A.6), the condition ${{A}_{{(\mathop p\limits^\circ )}}} = {{I}_{2}}$ leads to the equalities:

$${{\theta }_{{{{A}_{{(p)}}},\mathop p\limits^\circ }}} = 0{\kern 1pt} ,\;\;\;{{\vec {b}}_{{{{A}_{{(p)}}},\mathop p\limits^\circ }}} = 0{\kern 1pt} .$$

(A.8)

APPENDIX 2

1.1 PAULI–LUBANSKI PSEUDOVECTOR FOR FIELDS WITH AN ADDITIONAL VECTOR VARIABLE

The Poincaré group generators ${{\hat {P}}_{\nu }}$, ${{\hat {M}}_{{\nu \mu }}}$, acting in the field space $\Psi (p,\eta )$, have the form

$$\begin{gathered} {{{\hat {P}}}_{\mu }} = {{p}_{\mu }}{\kern 1pt} , \\ {{{\hat {M}}}_{{\mu \nu }}} = i\left( {{{p}_{\mu }}\frac{\partial }{{\partial {{p}^{\nu }}}} - {{p}_{\nu }}\frac{\partial }{{\partial {{p}^{\mu }}}} + {{\eta }_{\mu }}\frac{\partial }{{\partial {{\eta }^{\nu }}}} - {{\eta }_{\nu }}\frac{\partial }{{\partial {{\eta }^{\mu }}}}} \right){\kern 1pt} . \\ \end{gathered} $$

(B.1)

The Pauli–Lubanski pseudovector in the reference frame of a test momentum is defined as follows:

$${{\mathop W\limits^\circ }_{\mu }} = \frac{1}{2}{{\varepsilon }_{{\mu \nu \lambda \boldsymbol{\rho} }}}{{\hat {M}}^{{\lambda \boldsymbol{\rho} }}}\mathop {{{p}^{\nu }}}\limits^\circ i{{\varepsilon }_{{\mu \nu \lambda \boldsymbol{\rho} }}}\mathop {{{p}^{\nu }}}\limits^\circ {{\eta }^{\lambda }}\frac{\partial }{{\partial {{\eta }_{\boldsymbol{\rho} }}}}{\kern 1pt} .$$

(B.2)

Its components have the following form:

$$\begin{gathered} {{\mathop W\limits^\circ }_{0}} = {{\mathop W\limits^\circ }_{3}} = E\left( {\zeta \frac{\partial }{{\partial \zeta }} - \bar {\zeta }\frac{\partial }{{\partial \bar {\zeta }}}} \right){\kern 1pt} , \\ {{\mathop W\limits^\circ }_{{( + )}}} = - \frac{1}{2}\left( {{{{\mathop W\limits^\circ }}_{1}} + i{{{\mathop W\limits^\circ }}_{2}}} \right) = E\left( {{{\eta }^{ + }}\frac{\partial }{{\partial \zeta }} + \bar {\zeta }\frac{\partial }{{\partial {{\eta }^{ - }}}}} \right){\kern 1pt} , \\ {{\mathop W\limits^\circ }_{{( - )}}} = - \frac{1}{2}\left( {{{{\mathop W\limits^\circ }}_{1}} - i{{{\mathop W\limits^\circ }}_{2}}} \right) = E\left( {{{\eta }^{ + }}\frac{\partial }{{\partial \bar {\zeta }}} + \zeta \frac{\partial }{{\partial {{\eta }^{ - }}}}} \right){\kern 1pt} , \\ \end{gathered} $$

(B.3)

where the variables defined in (4.4) are used. It can be shown that in an arbitrary reference frame the components of the Pauli-Lubanski vector are given by the expressions:

$$\begin{array}{*{20}{c}} {{{W}_{0}}\, = \, - \frac{1}{2}{\kern 1pt} \left[ {({{\varepsilon }_{{( + )}}}\eta )\left( {{{\varepsilon }_{{( - )}}}\frac{\partial }{{\partial \eta }}} \right)\, - \,({{\varepsilon }_{{( - )}}}\eta )\left( {{{\varepsilon }_{{( + )}}}\frac{\partial }{{\partial \eta }}} \right)} \right]\, = \,{{W}_{3}}{\kern 1pt} ,} \\ {{{W}_{{( \pm )}}} = \left( {({{\varepsilon }_{{( \mp )}}}\eta )\left( {p\frac{\partial }{{\partial \eta }}} \right) - (p\eta )({{\varepsilon }_{{( \mp )}}}\frac{\partial }{{\partial \eta }})} \right).} \end{array}$$

(B.4)

APPENDIX 3

1.1 POLARIZATION VECTORS

Here we give expressions for the 4-polarization vectors used in the paper.

In the standard momentum frame, the 4-polarization vectors that are orthogonal to each other and to the massless test momentum $\mathop p\limits^\circ {\kern 1pt}$, have components

$${{({{\mathop \varepsilon \limits^\circ }_{{(1)}}})}_{\nu }} = (0,1,0,0){\kern 1pt} ,\quad {{({{\mathop \varepsilon \limits^\circ }_{{(2)}}})}_{\nu }} = (0,0,1,0){\kern 1pt} .$$

(C.1)

In an arbitrary basis, they are denoted by

$$\begin{array}{*{20}{c}} {{{\varepsilon }_{{(1)}}}: = \Lambda ({{A}_{{(p)}}}){\kern 1pt} {{{\mathop \varepsilon \limits^\circ }}_{{(1)}}}{\kern 1pt} ,\;\;\;{{\varepsilon }_{{(2)}}}: = \Lambda ({{A}_{{(p)}}}){\kern 1pt} {{{\mathop \varepsilon \limits^\circ }}_{{(2)}}}{\kern 1pt} .} \end{array}$$

(C.2)

The following linear combinations of vectors (C.2) are also used:

$${{\varepsilon }_{{( \pm )}}}: = {{\varepsilon }_{{(2)}}} \pm i{{\varepsilon }_{{(1)}}}{\kern 1pt} ,$$

(C.3)

which, by construction, satisfy the equations

$$p{{\varepsilon }_{{( \pm )}}} = 0,\quad {{\varepsilon }_{{( \pm )}}}{{\varepsilon }_{{( \pm )}}} = 0{\kern 1pt} ,\quad {{\varepsilon }_{{( + )}}}{{\varepsilon }_{{( - )}}} = - 2{\kern 1pt} .$$

(C.4)