Matched data for the regulated free-scalar Rosetta stone
Compare descriptions of one massive real scalar only after matching its system, state, boundary data, algebra and domain, normalization, regulator, and target two-point function. See the canonical table and its derivation.
System and symbols
- Mass, volume and retained modes
- The mass is m > 0 and the spatial torus is cubic with side L > 0 and periodic boundaries. N is a nonnegative integer cutoff on each component of the integer momentum label n = (n1, n2, n3): |ni| ≤ N for i = 1, 2, 3. It is not the number of retained modes. The allowed momenta are kn = 2πn/L.
- Modes, operators and state
- The positive frequency is ωn = √(|kn|2 + m2) > 0. The operators an and an† annihilate and create the corresponding modes; δnr is the Kronecker delta for two integer momentum labels. Use the positive-frequency Fock vacuum and the finite-particle subspace as a common invariant domain.
- Euclidean time discretization
- A declared interval [τi, τf], with τf > τi, has M ≥ 1 integer time steps of width Δτ = (τf − τi)/M. Match the time-boundary wave functions to prepare the same canonical state.
- Finite Gaussian kernel
- KM,N is the positive matrix of the finite Euclidean quadratic form with those boundary wave functions. Its inverse KM,N−1 is the exact covariance of that regulated Gaussian. The canonical comparison must use the same M-step Trotter kernel, with matching ordering and normalization.
| Rosetta entry | Matched reference data | Appearance across descriptions | What may change or must still be proved |
|---|---|---|---|
| System | Massive real scalar on ℝ × 𝕋L3, with 𝕋L3 the cubic torus of side L | The finite canonical and functional descriptions use the same regulated system. Later Lorentzian, Euclidean, spectral, and structural statements concern the corresponding family only after the limits required by each statement are controlled. | Massless, complex, spinor, and vector fields require separate checks. |
| State and boundary data | Positive-frequency Fock vacuum, periodic space, and matched time-boundary wave functions | Canonical vacuum expectations, Lorentzian boundary values, and the Euclidean covariance are compared for the same prepared Gaussian state. | Thermal, in–in, scattering, and curved-spacetime states are different data. |
| Algebra and domain | Canonical commutation relation algebra for finitely many modes, represented on Fock space with the finite-particle subspace as a common invariant domain | Canonical operators and finite Gaussian variables provide two regulated descriptions; the spectral and structural statements use the resulting two-point object. | Continuum fields are distributions and need domain and product control. |
| Fourier and mode normalization | L−3/2 modes, 1/√(2ωn), and [an,ar†] = δnr | The same normalization fixes the canonical commutator, finite-volume propagator, and source covariance; after the continuum limit, retaining canonical normalization fixes the free spectral pole weight. | Another Fourier convention is equivalent only after every measure, delta function, mode, operator, and state normalization is translated together. |
| Regulator and limits | Finite spatial mode set and an M-step Trotter discretization on a declared Euclidean time interval | The finite-mode canonical sums and the covariance KM,N−1 of the finite Gaussian discretization are exact within their respective regulated constructions. They agree only when KM,N is built from the same M-step Trotter kernel, not from exact-time evolution. | No time-continuum, cutoff-removal, infinite-volume, or zero-mass limit is automatic. |
| Target two-point object | Vacuum time-ordered two-point function and its Euclidean covariance | Canonical ordering, Gaussian source differentiation, spectral support, and continuation are compared for this declared two-point object. A correlator is not automatically a directly measured observable. | Responses, in–in expectations, unordered correlators, and scattering amplitudes use different orderings, contours, or asymptotic data. |
| Strength of comparison | Exact Gaussian identities and equality with the same Trotterized kernel | The finite canonical and functional descriptions agree after the system, state, boundary data, normalization, regulator, and target two-point object are matched. | Exact-time evolution, regulator removal, analytic continuation, reconstruction, and framework equivalence remain separate conditional claims. |
Scope of the comparison
Equality with a matched finite Trotter construction is distinct from equality with exact canonical time evolution, which additionally requires the time-step limit. Spatial cutoff removal and the infinite-volume limit are separate. Continuum causal support, exact Poincaré covariance, the continuum delta and the spectral representation require their corresponding controlled limits and hypotheses. A finite cubic cutoff is not Lorentz invariant and need not preserve exact continuum microcausality. Analytic continuation, reconstruction and universal equivalence of frameworks do not follow from a finite Gaussian identity.