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Finite-mode Dirac first-order constraint reduction

Statement

Assume Countable Choice, the constants in Charged gauge data and SI-natural conventions, $ℏ > 0$ , and a finite integer $N \geq 1$ . Let the real configuration space be $Q = R^{2 N}$ with coordinates $(x_{1}, y_{1}, \dots, x_{N}, y_{N})$ and let $T^{*} Q ≅ R^{4 N}$ have corresponding real canonical momentum coordinates $(P_{1}^{x}, P_{1}^{y}, \dots, P_{N}^{x}, P_{N}^{y})$ . On real smooth functions define the canonical Poisson bracket ${F, G} = \sum_{j = 1}^{N} (\frac{\partial F}{\partial x_{j}} \frac{\partial G}{\partial P_{j}^{x}} - \frac{\partial F}{\partial P_{j}^{x}} \frac{\partial G}{\partial x_{j}} + \frac{\partial F}{\partial y_{j}} \frac{\partial G}{\partial P_{j}^{y}} - \frac{\partial F}{\partial P_{j}^{y}} \frac{\partial G}{\partial y_{j}}),$ and extend it $C$ -bilinearly to complex-valued smooth functions on $T^{*} Q$ . Define the complex coordinate notation $z_{j} = x_{j} + i y_{j}, {\overset{ˉ}{z}}_{j} = x_{j} - i y_{j}, p_{j} = \frac{P_{j}^{x} - i P_{j}^{y}}{2}, {\overset{ˉ}{p}}_{j} = \frac{P_{j}^{x} + i P_{j}^{y}}{2} .$ These definitions give ${z_{j}, p_{k}} = δ_{j k}, {{\overset{ˉ}{z}}_{j}, {\overset{ˉ}{p}}_{k}} = δ_{j k},$ with all other elementary brackets zero; the bracket is extended by bilinearity and the Leibniz rule. The first-order kinetic term $\frac{i ℏ}{2} \sum_{j} ({\overset{ˉ}{z}}_{j} {\dot{z}}_{j} - {\dot{\overset{ˉ}{z}}}_{j} z_{j})$ gives primary constraints $C_{j} = p_{j} - \frac{i ℏ}{2} {\overset{ˉ}{z}}_{j}, {\overset{ˉ}{C}}_{j} = {\overset{ˉ}{p}}_{j} + \frac{i ℏ}{2} z_{j} .$ In the order $(C_{1}, \dots, C_{N}, {\overset{ˉ}{C}}_{1}, \dots, {\overset{ˉ}{C}}_{N})$ their constraint matrix and inverse are $M = (\begin{array}{cc} 0 & - i ℏ I_{N} \\ i ℏ I_{N} & 0 \end{array}), M^{- 1} = (\begin{array}{cc} 0 & - i I_{N} / ℏ \\ i I_{N} / ℏ & 0 \end{array}) .$ Here, “second class” means that the constraint matrix $M_{A B} = {χ_{A}, χ_{B}}$ is invertible. Thus these constraints are second class. For a real smooth finite-mode Hamiltonian $H \in C^{\infty} (Q; R)$ , viewed as independent of the momenta, define the Wirtinger derivative $\frac{\partial H}{\partial {\overset{ˉ}{z}}_{j}} : = \frac{1}{2} (\frac{\partial H}{\partial x_{j}} + i \frac{\partial H}{\partial y_{j}}) .$ The Dirac bracket ${F, G}_{D} = {F, G} - {F, χ_{A}} (M^{- 1})_{A B} {χ_{B}, G}, χ = (C_{1}, \dots, C_{N}, {\overset{ˉ}{C}}_{1}, \dots, {\overset{ˉ}{C}}_{N}),$ satisfies ${z_{j}, {\overset{ˉ}{z}}_{k}}_{D} = - i δ_{j k} / ℏ$ and ${z_{j}, z_{k}}_{D} = {{\overset{ˉ}{z}}_{j}, {\overset{ˉ}{z}}_{k}}_{D} = 0$ . For a real smooth finite-mode Hamiltonian $H (z, \overset{ˉ}{z})$ the reduced equation is $i ℏ {\dot{z}}_{j} = \partial H / \partial {\overset{ˉ}{z}}_{j}$ . These are ordinary commuting coordinates; they do not derive CAR by canonical quantization.

Facts & Assumptions

Given: Countable Choice, the charged normalization conventions, $ℏ > 0$ , a finite mode number $N \geq 1$ , and the canonical brackets defined in the statement.

[A1]

The natural convention uses positive $ℏ$ and a first-order covariant Dirac kinetic term. Charged gauge data and SI-natural conventions

[A2]

Assume Countable Choice. The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F1]

The natural covariant derivative is $D_{a} = \partial_{a} - i g a_{a}$ and $ℏ > 0$ . Charged gauge data and SI-natural conventions

Proof

Given: [A1], [A2], $ℏ > 0$ , and the canonical brackets in the statement.

1.1A1A2F1givenalgebra

Under the explicit Countable Choice assumption [A2], write the displayed kinetic term in the underlying real coordinates: $\frac{i ℏ}{2} \sum_{j} ({\overset{ˉ}{z}}_{j} {\dot{z}}_{j} - {\dot{\overset{ˉ}{z}}}_{j} z_{j}) = ℏ \sum_{j} (y_{j} {\dot{x}}_{j} - x_{j} {\dot{y}}_{j}) .$ Its real canonical momenta are therefore $P_{j}^{x} = ℏ y_{j}$ and $P_{j}^{y} = - ℏ x_{j}$ . By the complex momentum definitions in the Statement, $p_{j} = i ℏ {\overset{ˉ}{z}}_{j} / 2$ and ${\overset{ˉ}{p}}_{j} = - i ℏ z_{j} / 2$ . Moving these velocity-independent expressions to the left gives exactly $C_{j} = 0$ and ${\overset{ˉ}{C}}_{j} = 0$ . The finite canonical calculation uses the positive $ℏ$ fixed in [F1].

2.1A1F1step 1.1givenalgebra

For each $j, k$ , bilinearity and the elementary brackets give ${C_{j}, C_{k}} = 0, {{\overset{ˉ}{C}}_{j}, {\overset{ˉ}{C}}_{k}} = 0, {C_{j}, {\overset{ˉ}{C}}_{k}} = {p_{j}, (i ℏ / 2) z_{k}} + {- (i ℏ / 2) {\overset{ˉ}{z}}_{j}, {\overset{ˉ}{p}}_{k}} = - \frac{i ℏ}{2} δ_{j k} - \frac{i ℏ}{2} δ_{j k} = - i ℏ δ_{j k} .$ Antisymmetry gives ${{\overset{ˉ}{C}}_{j}, C_{k}} = i ℏ δ_{j k}$ , which is exactly the displayed block matrix $M$ .

3.1F1step 2.1givenalgebra

Multiplication of the two displayed block matrices gives $M M^{- 1} = M^{- 1} M = I_{2 N}$ ; each off-diagonal product is $(- i ℏ) (i / ℏ) = 1$ or $(i ℏ) (- i / ℏ) = 1$ , and the diagonal blocks are zero. Because $ℏ > 0$ , $M$ is invertible, so these primary constraints are second class by definition.

4.1step 3.1givenalgebra

The canonical coordinate bracket ${z_{j}, {\overset{ˉ}{z}}_{k}}$ is zero. In the Dirac correction, the only nonzero factors are ${z_{j}, C_{ℓ}} = δ_{j ℓ}$ and ${{\overset{ˉ}{C}}_{m}, {\overset{ˉ}{z}}_{k}} = - δ_{m k}$ . Hence ${z_{j}, {\overset{ˉ}{z}}_{k}}_{D} = 0 - δ_{j ℓ} (M^{- 1})_{C_{ℓ}, {\overset{ˉ}{C}}_{m}} (- δ_{m k}) = (M^{- 1})_{C_{j}, {\overset{ˉ}{C}}_{k}} = - \frac{i}{ℏ} δ_{j k} .$ The same formula has no nonzero block for ${z_{j}, z_{k}}_{D}$ or ${{\overset{ˉ}{z}}_{j}, {\overset{ˉ}{z}}_{k}}_{D}$ , so both vanish.

5.1F1step 4.1givenalgebra∎

For a smooth $H (z, \overset{ˉ}{z})$ , the only nonzero reduced coordinate bracket in ${z_{j}, H}_{D}$ is with ${\overset{ˉ}{z}}_{j}$ , so step 4.1 gives ${\dot{z}}_{j} = {z_{j}, H}_{D} = - i ℏ^{- 1} \partial H / \partial {\overset{ˉ}{z}}_{j}$ . Multiplying by $i ℏ$ proves the stated equation. The calculation remains finite-dimensional for every $N$ and introduces no infinite-dimensional Poisson bracket or fermionic anticommutator.

Finite-mode Dirac first-order constraint reduction

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources