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Two-pi spin rotation and even-observable invariance

Physical scope

One exact 2π or 4π spatial rotation in the selected free Dirac CAR model. The claim concerns the norm-closed bounded even algebra generated by smeared fields and contains no experiment, point-field product or general spin-statistics theorem.

Example

Define $A_{D}^{e v e n}$ as the norm closure of finite linear combinations of the identity and even-degree products of $Ψ (F)$ and $Ψ (F)^{†}$ , for Schwartz spinor tests $F$ . In the selected free Dirac model, let $U_{2 π}$ implement a spatial rotation by $2 π$ and let $U_{4 π}$ implement a rotation by $4 π$ . On the $n$ -particle sector of the adopted antisymmetric Fock decomposition, indexed by $n \in N_{0} = {0, 1, 2, \dots}$ , $U_{2 π}$ acts as $(- 1)^{n}$ times the identity; $U_{4 π} = I$ . Every finite even polynomial in bounded spacetime-smeared Dirac fields and their adjoints is invariant under conjugation by $U_{2 π}$ , and therefore so is $A_{D}^{e v e n}$ . This is an exact model calculation; it does not infer a general spin-statistics theorem.

Facts & Assumptions

Given: The physical model and mathematical conventions of Dirac CAR, covariance, graded locality and positive Fock energy.

[F1]

Dirac CAR, covariance, graded locality and positive Fock energy proves that the $2 π$ spin matrix is $- I_{4}$ , the test pullback sends $F$ to $- F$ , the field and its adjoint change sign, $U_{2 π}$ acts on the $n$ -particle sector as $(- 1)^{n}$ for every $n \in N_{0}$ , and the $4 π$ action is the identity.

[F2]

Dirac CAR, covariance, graded locality and positive Fock energy supplies the strongly continuous unitary Fock representation $U$ and its covariance action on the smeared field.

[F3]

Dirac CAR, covariance, graded locality and positive Fock energy states that the bounded smeared field $Ψ$ is antilinear in every Schwartz spinor test $F$ .

Verification

1.1F1F2F3givenalgebra

By [F1] and the covariance action in [F2], $U_{2 π} Ψ (F) U_{2 π}^{- 1} = Ψ (- F)$ . Antilinearity from [F3] gives $Ψ (- F) = - Ψ (F)$ ; taking adjoints gives the same sign for $Ψ (F)^{†}$ . The $4 π$ action is the identity by [F1].

2.1step 1.1algebra

If $M = A_{1} \dots A_{r}$ is a monomial whose factors are smeared fields or their adjoints, repeated use of step 1.1 and multiplicativity of unitary conjugation gives $U_{2 π} M U_{2 π}^{- 1} = (- 1)^{r} M$ . Thus every even-degree monomial is fixed. Products of even-degree monomials and adjoints of even-degree monomials still have even degree, so finite linear combinations of them and the identity form a unital *-algebra; linearity fixes every such polynomial.

3.1F2F3step 2.1algebra∎

Conjugation by the unitary in [F2] is norm-continuous on bounded operators: $∥ U O U^{- 1} - U O^{'} U^{- 1} ∥ = ∥ O - O^{'} ∥$ . Therefore every norm limit of finite even polynomials is also invariant. The conclusion concerns the algebra defined in the Example; it does not assert that pointwise field products exist.

Depends on

Dirac CAR, covariance, graded locality and positive Fock energy

Two-pi spin rotation and even-observable invariance

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources