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Gauge phase periodicity, a linear gauge, and the dimension vector of q chi over hbar

Example

Assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )). Fix $ℏ > 0$ , $m > 0$ , $q \in R$ and consider the gauge multiplication $G : = M_{e^{i q χ / ℏ}}$ of Gauge multiplication preserves H2 and transports the magnetic momentum.

(a) Periodicity of the phase. If $χ, χ^{'} : R^{3} \to R$ are real functions and $χ - χ^{'}$ takes values in $\frac{2 π ℏ}{q} Z$ when $q \neq 0$ , then $e^{i q χ / ℏ} = e^{i q χ^{'} / ℏ}$ pointwise; in particular $M_{e^{i q (χ + 2 π ℏ / q) / ℏ}} = M_{e^{i q χ / ℏ}}$ . If $q = 0$ then $G = I$ .

(b) A linear gauge. Let $A = 0$ , $Φ = 0$ , let $c \in R^{3}$ and let $χ (x) : = c \cdot x$ , time-independent. Then $A_{j}^{'} = \partial_{j} χ = c_{j}$ , $Φ^{'} = - \partial_{t} χ = 0$ , and on $W^{2, 2} (R^{3}; C)$ $K_{c, 0} = G K_{0, 0} G^{- 1}, G (P_{j}) G^{- 1} = P_{j} - q c_{j}, e^{- i t K_{c, 0} / ℏ} = G e^{- i t K_{0, 0} / ℏ} G^{- 1},$ so the constant vector potential is removed by the unitarity $G$ ; in particular, $K_{c, 0}$ and $K_{0, 0}$ have the same spectrum, and their operator-domain pairings agree under $G$ : $⟨ K_{c, 0} G f, G f ⟩ = ⟨ K_{0, 0} f, f ⟩ (f \in W^{2, 2} (R^{3}; C)) .$

(c) Dimension vector. In the SI dimension convention with base dimensions $(m, k g, s, A)$ and dimension vectors $[q] = (0, 0, 1, 1), [χ] = (2, 1, - 2, - 1), [ℏ] = (2, 1, - 1, 0),$ the combination $q χ / ℏ$ has dimension vector $[q] + [χ] - [ℏ] = (0, 0, 0, 0)$ : it is dimensionless in that convention. (The dimension-vector formalism is a bookkeeping device; the assignment of dimension vectors to the physical quantities is an external convention, not a mathematical consequence.)

Verification

Given: $ℏ > 0$ , $m > 0$ , $q \in R$ ; gauge multipliers as in Gauge multiplication preserves H2 and transports the magnetic momentum; in (b) the data $A = 0$ , $Φ = 0$ , $χ (x) = c \cdot x$ with $χ$ regarded as a function on $R^{3} \times R$ constant in $t$ .

Proof technique: direct verification of the phase algebra, of the conjugation identities and of the finite dimension-vector arithmetic.

[F1] For real $α$ , Euler's formula gives $∣ e^{i α} ∣ = 1$ ( $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ ); the complex exponential's fibres are exactly its translates by $2 π i Z$ ( $\ker (\exp) = 2 π i Z$ , and $\exp z = \exp w$ exactly when $z - w \in 2 π i Z$ ).

[F2] The covariance identity of Gauge multiplication preserves H2 and transports the magnetic momentum clause (c): $(P_{j} - q (A_{j} + \partial_{j} χ)) G = G (P_{j} - q A_{j})$ on $W^{1, 2}$ ; the gauge unitarity and $W^{2, 2}$ -preservation of $G$ are clauses (a)-(b) there.

[F3] $K_{A, Φ}$ of Bounded-coefficient magnetic Schrodinger operator is self-adjoint on $W^{2, 2}$ and equals the displayed magnetic square plus scalar potential (Self-adjointness of the bounded-coefficient magnetic Schrodinger operator); $P_{j}$ is linear on its domain.

1.1givenF1algebra

[Periodicity] If $q \neq 0$ and $χ (x) - χ^{'} (x) = \frac{2 π ℏ}{q} k (x)$ with $k (x) \in Z$ , then $i \frac{q}{ℏ} (χ - χ^{'}) (x) = 2 π i k (x) \in 2 π i Z$ . By [F1], $e^{i \frac{q}{ℏ} χ (x)} = e^{i \frac{q}{ℏ} χ^{'} (x)}$ pointwise, hence the multipliers agree; $χ + 2 π ℏ / q$ is the case $χ^{'} = χ$ . If $q = 0$ then $e^{i 0 \cdot χ / ℏ} = 1$ . This is (a).

1.2givenF2F3algebra

[Linear gauge conjugations] Take $χ (x) = c \cdot x$ , $\partial_{j} χ = c_{j}$ constant and $\partial_{t} χ = 0$ . By [F2] with $A = 0$ , $(P_{j} - q c_{j}) G = G P_{j}$ on $W^{1, 2}$ . For the inverse, compute on $f \in W^{1, 2}$ : $D_{j} (G^{- 1} f) = G^{- 1} (D_{j} f - i \frac{q c_{j}}{ℏ} f)$ , hence $P_{j} G^{- 1} f = G^{- 1} (P_{j} - q c_{j}) f$ ; multiplying by $G$ on the left gives $G P_{j} G^{- 1} = P_{j} - q c_{j}$ on $W^{1, 2}$ .

2.1givenstep 1.2F3algebra

[Operator identity and propagator] For $u \in W^{2, 2}$ , $(P_{j} - q c_{j})^{2} G u = (P_{j} - q c_{j}) G (P_{j} u) = G P_{j} (P_{j} u)$ by two uses of step 1.2, so $K_{c, 0} G = G K_{0, 0}$ on $W^{2, 2}$ ; since $G$ is a bijection of $W^{2, 2}$ onto itself ([F2]), this gives $K_{c, 0} = G K_{0, 0} G^{- 1}$ with common domain $W^{2, 2}$ . Both operators are self-adjoint by [F3]; the conjugate of the spectral PVM of $K_{0, 0}$ by $G$ is a regular PVM representing $K_{c, 0}$ , hence is its spectral PVM by the uniqueness clause of the spectral theorem, so every bounded Borel function satisfies $h (K_{c, 0}) = G h (K_{0, 0}) G^{- 1}$ and in particular $e^{- i t K_{c, 0} / ℏ} = G e^{- i t K_{0, 0} / ℏ} G^{- 1}$ (Spectral theorem for unbounded self-adjoint operators (PVM form), A self-adjoint operator generates a strongly continuous unitary group, Integral of a measurable function against a projection-valued measure); the spectra coincide because $G$ carries the resolvent of $K_{0, 0}$ to that of $K_{c, 0}$ , $(K_{c, 0} - z)^{- 1} = G (K_{0, 0} - z)^{- 1} G^{- 1}$ . This is (b).

3.1givenalgebra∎

[Dimension vector] With the stated vectors, $[q] + [χ] - [ℏ] = (0 + 2 - 2, 0 + 1 - 1, 1 - 2 + 1, 1 - 1 - 0) = (0, 0, 0, 0)$ , so the combination $q χ / ℏ$ has zero dimension vector; this is a four-component integer computation. This is (c).

Gauge phase periodicity, a linear gauge, and the dimension vector of q chi over hbar

Example

Verification

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