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Normalized Fourier spectrum on a flat Kaluza–Klein circle

Physical scope

Exact quadratic spectrum of a free real scalar on flat $(d + 1)$ -dimensional Minkowski spacetime with one compact spacelike circle. The external field configurations have compact support. No interacting truncation, string winding calculation, warping, or empirical claim is included.

Example

Let $M_{d} = R^{1, d - 1}$ have the flat Minkowski metric $η_{d} = diag (-, +, \dots, +)$ , let $D = d + 1$ , and let $y \sim y + 2 π R$ be a flat spacelike circle of radius $R > 0$ . Take a real free scalar $Φ \in C_{c}^{\infty} (M_{d} \times S_{R}^{1}, R)$ with nonnegative mass parameter $m_{0} \geq 0$ and action $S = - \frac{1}{2} \int_{R^{1, d - 1}} d^{d} x \int_{0}^{2 π R} d y (η^{μ ν} \partial_{μ} Φ \partial_{ν} Φ + (\partial_{y} Φ)^{2} + m_{0}^{2} Φ^{2}),$ in units $ℏ = c = 1$ . Here $[x^{μ}] = [y] = [R] = l e n g t h$ , $[m_{0}] = (l e n g t h)^{- 1}$ , $[Φ] = (l e n g t h)^{- (D - 2) / 2} = (l e n g t h)^{- (d - 1) / 2}$ , and the normalized basis $u_{n} (y) = (2 π R)^{- 1 / 2} e^{i n y / R}$ has dimension $(l e n g t h)^{- 1 / 2}$ . The mode coefficient $ϕ_{n}$ therefore has dimension $(l e n g t h)^{- (d - 2) / 2}$ , making $S$ dimensionless. Set $Φ (x, y) = \sum_{n \in Z} ϕ_{n} (x) u_{n} (y), ϕ_{n} (x) = \int_{0}^{2 π R} d y u_{n} (y)^{*} Φ (x, y) .$ Then $\int_{0}^{2 π R} u_{n}^{*} u_{m} d y = δ_{n m}$ , reality gives $ϕ_{- n} = \overline{ϕ_{n}}$ , and the quadratic action is $S = - \frac{1}{2} \int_{R^{1, d - 1}} d^{d} x \sum_{n \in Z} (η^{μ ν} \partial_{μ} ϕ_{n} \partial_{ν} ϕ_{- n} + (m_{0}^{2} + n^{2} / R^{2}) ϕ_{n} ϕ_{- n}) .$ Thus the free mode masses obey $m_{n}^{2} = m_{0}^{2} + n^{2} / R^{2}$ . The $n$ and $- n$ coefficients are conjugates and represent two real fields for each $n > 0$ , not two independent complex scalars. This exact quadratic calculation does not assert that discarding nonzero modes is an interacting consistent truncation or that field momentum modes include string winding states.

Example references

Kaluza–Klein product and warped reduction data

Absolutely convergent Fourier reconstruction for C2 periodic functions

Facts & Assumptions

Given: Fixed unwarped product $M_{d} \times S_{R}^{1}$ , $R > 0$ , and the real scalar action displayed in the statement.

[F1]

Kaluza–Klein product and warped reduction data: a product metric and coordinate period are declared ansatz data. The particular flat circle here is a model specialization.

[F2]

Absolutely convergent Fourier reconstruction for C2 periodic functions: for every $C^{2}$ $2 π$ -periodic function, normalized Fourier exponentials are orthonormal, the Fourier series reconstructs it absolutely and uniformly, and the symmetric derivative partial sums converge in normalized $L^{2}$ . Applied to $f_{x} (θ) = Φ (x, R θ)$ and its external derivatives, this justifies the mode expansion and the quadratic circle integrals.

[A1]

The field and all its external derivatives are smooth and compactly supported in $x$ , so the external action integral is finite. For each fixed $x$ , the field and its external derivatives are smooth periodic functions of $y$ ; the Fourier lemma's $C^{2}$ and $L^{2}$ conclusions justify the circle expansions and quadratic integrals. Circle integrations are exact and have no boundary term by periodicity. The mass-squared parameter satisfies $m_{0}^{2} \geq 0$ .

Verification

1.1givenF2A1algebra

Rescale $θ = y / R$ . Then $f_{x} (θ) = Φ (x, R θ)$ is $2 π$ -periodic, and its Fourier coefficient is $c_{n} (x) = \frac{1}{2 π} \int_{0}^{2 π} f_{x} (θ) e^{- i n θ} d θ = \frac{1}{2 π R} \int_{0}^{2 π R} Φ (x, y) e^{- i n y / R} d y = \frac{ϕ_{n} (x)}{\sqrt{2 π R}} .$ Thus [F2] yields precisely $Φ = (2 π R)^{- 1 / 2} \sum_{n} ϕ_{n} e^{i n y / R}$ . Also, for integers $n, m$ , direct integration gives $\int_{0}^{2 π R} u_{n}^{*} u_{m} d y = \frac{1}{2 π R} \int_{0}^{2 π R} e^{i (m - n) y / R} d y = δ_{n m},$ because the integral is $2 π R$ when $m = n$ and otherwise its antiderivative has equal endpoint values.

2.1givenF2A1step 1.1algebra

Since $Φ$ is real, conjugating its coefficient integral gives $\overline{ϕ_{n}} = \int u_{n} Φ d y = ϕ_{- n}$ . The same orthogonality gives the nonconjugated identity $\int u_{n} u_{m} d y = δ_{n, - m}$ . Substituting the Fourier expansion into the external kinetic and mass terms therefore pairs modes $n$ and $- n$ as $\partial ϕ_{n} \partial ϕ_{- n}$ and $m_{0}^{2} ϕ_{n} ϕ_{- n}$ .

3.1givenA1step 2.1algebra

Differentiating $u_{n}$ gives $\partial_{y} u_{n} = (i n / R) u_{n}$ . Hence $\int_{0}^{2 π R} (\partial_{y} Φ)^{2} d y = \sum_{n, m} ϕ_{n} ϕ_{m} \frac{i n}{R} \frac{i m}{R} \int_{0}^{2 π R} u_{n} u_{m} d y = \sum_{n} \frac{n^{2}}{R^{2}} ϕ_{n} ϕ_{- n},$ where $m = - n$ in each surviving term. This proves the stated action and mass formula.

4.1givenA1step 2.1step 3.1algebra

For $n > 0$ write $ϕ_{n} = (a_{n} + i b_{n}) / \sqrt{2}$ and $ϕ_{- n} = (a_{n} - i b_{n}) / \sqrt{2}$ with real $a_{n}, b_{n}$ . The two terms indexed by $n$ and $- n$ in the full sum, including its overall $1 / 2$ , combine to the canonical kinetic and mass terms for $a_{n}$ and $b_{n}$ once; the zero mode $ϕ_{0}$ is real and appears once. This verifies that the complex sum does not double-count independent degrees of freedom.

5.1givenF1A1step 3.1algebra∎

The calculation used only the free quadratic operator. A nonlinear interaction can multiply retained harmonics and source omitted harmonics, while string winding states are separate string sectors rather than Fourier components of this scalar field. Therefore the mass eigenvalues alone imply neither consistent truncation nor a winding spectrum.

Normalized Fourier spectrum on a flat Kaluza–Klein circle

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources