The vacuum and outer-time reconstruction

Why the top eigenvector is the vacuum

The normalized transfer operator T advances one outer Euclidean time step. Its eigenvalues lie in (0,1], with zero allowed only as an accumulation point. The physical Hamiltonian H=−a_t⁻¹log T sends the largest transfer eigenvalue to the smallest energy. Normalizing that eigenvalue to one fixes the ground energy to zero.

The strictly positive integral kernel supplies more than nonnegative eigenvalues. It sends every nonzero nonnegative function to a strictly positive function. The compact positive-kernel theorem then gives a simple top eigenvalue and an everywhere positive normalized eigenfunction Ω.

The physical excitation space is Ω^⊥. A gap statement concerns the spectrum there, because the vacuum has zero energy by construction. Subtracting a vacuum expectation from an observable produces a vector in this space: ψ=(A−⟨Ω,AΩ⟩I)Ω.

At a fixed regulator, smoothness and compactness also imply a positive minimum of Ω. This guarantees that every open configuration region of positive Haar measure carries nonzero vacuum probability. It is exactly why a nontrivial sharp multiplication window normally divides the vacuum between its two sides.

Source: Vacuum obstructions and form Feshbach reduction, Theorem 3.3 and §5.5, pp.8–9,14–15.

Turn the vacuum into a probability measure

Let μ be product Haar measure on the slice configuration space. Define dν=Ω²dμ. Since Ω is normalized, ν is a probability measure. Multiplication by Ω defines a unitary map U_Ω:L²(ν)^gauge→L²(μ)^gauge, because ∥Ωf∥²_μ=∥f∥²_ν.

Conjugating the transfer operator gives T̂=U_Ω⁻¹TU_Ω. If t(X,Y) is the original integral kernel relative to μ, then the transformed kernel relative to ν is

The second identity follows by substituting TΩ=Ω. Thus T̂ preserves constants and is a Markov operator. It remains self-adjoint, so the probability process is reversible: ν(dX)t̂(X,Y)ν(dY) is symmetric under exchanging X and Y.

This transformation changes the representation, not the spectrum. The vacuum becomes the constant function one, and vacuum-orthogonal vectors become mean-zero functions. Quantum energy estimates can therefore be expressed as variance or Dirichlet-form inequalities on the vacuum probability space.

The measure ν is generally not Haar measure and not the classical Gibbs weight e^−Vdμ. It is determined by the full kinetic-plus-potential transfer problem through Ω.

Source: Vacuum obstructions and form Feshbach reduction, §4.1, p.10.

Construct a two-sided vacuum process

The probability kernel T̂ and invariant measure ν determine consistent finite-time distributions. For ordered integer times, multiply the appropriate transition kernels and integrate against ν at the first time. Consistency under marginalization follows because the kernels preserve constants.

The resulting process is stationary on all integer outer times. Positive and negative times are treated symmetrically by reversibility. This is a direct vacuum construction: the one-time distribution is ν and the correlations are determined by the ground-state-transformed transfer operator.

For bounded gauge-invariant multiplication observables A and B separated by k≥0 steps, the quantum and probability descriptions agree:

Centered correlations therefore probe the excited spectrum. In a finite two-state example with a single nonvacuum transfer eigenvalue t∈(0,1), a normalized centered eigenobservable has correlation t^k. Writing physical time s=ka_t gives t^k=e^−sΔ, with Δ=−log t/a_t.

This example illustrates how an exponential correlation scale and a Hamiltonian energy are related. It does not supply a uniform value for Δ in a changing lattice gauge regulator.

Source: Vacuum obstructions and form Feshbach reduction, Theorem 4.2 and Proposition 3.5, pp.9–10.

Reflection positivity uses outer time

Let θ reflect integer outer time, so (θX)_k=X_−k. For a complex cylinder observable F depending only on times 0,1,…,r, define the reflected antilinear operation ΘF=overline{F∘θ}. Conditioned on the configuration at time zero, the future and past are independent.

Reversibility identifies the reflected past conditional law with the future conditional law. Therefore

The nonnegative square is the finite-regulator reflection-positivity argument. It does not require interpreting the auxiliary cycle as a causal or Lorentzian time direction. Reflection occurs across an outer-time slice.

Osterwalder–Schrader reconstruction takes positive-time observables, forms this positive semidefinite inner product, and quotients out vectors of zero norm. Here the quotient has an especially concrete description: F is identified with its conditional expectation given X₀. Bounded gauge-invariant functions of X₀ are already present and dense.

The reconstructed Hilbert space is therefore naturally L²(ν)^gauge. One-step translation is T̂, and its generator is unitarily equivalent to H. These exact finite-regulator statements do not supply the additional regularity, rotation covariance and locality required of a continuum relativistic quantum field theory.

Source: Vacuum obstructions and form Feshbach reduction, Theorem 4.2, p.10.

A periodic trace state is a different preparation

A finite periodic outer-time lattice of M steps closes the history into a loop. Its normalized expectations involve Tr(T^MA)/Tr(T^M). In the Hamiltonian representation this is a thermal trace with inverse temperature Ma_t, in units where the thermal and action constants have been fixed.

At finite M, this trace includes excited eigenvectors. The pure vacuum expectation is ⟨Ω,AΩ⟩. They coincide only in special cases or after a suitable limiting procedure.

At fixed regulator, the simple top eigenvalue and trace-class property make the normalized expression converge to the vacuum expectation as M→∞. If volume or cutoff is also changing, uniform control is needed to exchange these limits or replace them by a diagonal limit.

This distinction matters whenever a simulation uses periodic time. A positive finite-temperature transfer construction is valuable evidence about the regulated theory, but it should report its time extent and excited-state contamination before being described as a vacuum calculation. The two-sided process avoids this ambiguity by beginning with the exact ground-state measure.

Source: Vacuum obstructions and form Feshbach reduction, Remark 4.3, p.11.

The vacuum is neutral under the auxiliary shift

Let R_τ shift all link variables by one auxiliary step. In the frozen regulator it commutes with the kinetic operator, potential, transfer operator and mixed-energy multiplication operator. Hence R_τΩ is another normalized positive top eigenvector.

Vacuum uniqueness forces R_τΩ=Ω. The average P₀=N⁻¹Σ_j=0^N−1R_τ^j is the orthogonal projection onto the neutral sector, and that sector reduces H and T.

Neutrality means invariance under a simultaneous cyclic relabelling. It does not mean independence of the auxiliary variables. A trace of auxiliary holonomy can be invariant under changing the starting point of the cycle; averages of relative auxiliary observables can also be neutral.

Consequently, restricting to the neutral sector is only a candidate physical projection. A four-dimensional limit must show that centered neutral auxiliary observables lose all matrix elements between physical-core states, or are otherwise absorbed into the declared physical theory. The vacuum’s symmetry does not prove that disappearance.

This is the same general reason that the absence of net charge does not imply the absence of dynamics. A neutral subspace can contain many nonvacuum states and nontrivial correlations.

Source: Vacuum obstructions and form Feshbach reduction, §4.2, p.11.

A physical gap becomes a variance inequality

Because Ω corresponds to the constant function, a Hamiltonian gap Δ is equivalent to a transfer contraction on mean-zero functions. The equivalent probability statement is

The left side is a discrete-time Dirichlet form. For a symmetric Markov kernel it is also one half of the integral of |f(X)−f(Y)|² against the stationary one-step pair distribution. It measures the average change of f under one step.

The factor 1−e^−a_tΔ is approximately a_tΔ when the time step is small. A uniform physical gap therefore corresponds to a one-step variance coefficient that usually tends to zero linearly in a_t. Requiring that coefficient to stay fixed would demand a diverging energy scale.

The source proves the abstract equivalence. Producing one Δ valid for every relevant regulator and spatial volume is the remaining multiscale Poincaré task. A local inequality or a finite-regulator numerical gap does not automatically provide its constants.

The kernel form follows directly by expanding the square. Stationarity makes the integrals of |f(X)|² and |f(Y)|² equal, and reversibility makes the cross term real after symmetrization. Thus ⟨f,(I−T̂)f⟩=(1/2)∫|f(X)−f(Y)|²t̂(X,Y)dν(X)dν(Y). Constants contribute zero to both sides. A global Poincaré estimate says that every other fluctuation is detected by this one-step variation with the stated coefficient. It must hold for all functions in the relevant dense form domain, including fluctuations spread across many blocks; checking only a chosen local observable would leave other slowly varying modes uncontrolled.

Source: Auxiliary-time lattice gauge framework, §26, printed pp.55–56.

What must survive reconstruction in a limit

The finite-regulator construction gives a well-defined vacuum theory at each n. To obtain a continuum theory, specify a common algebra of renormalized observables and prove convergence of its correlations. Reflection positivity passes through limits of its finite quadratic forms, but the rest of the continuum structure requires additional arguments.

In particular, the Hamiltonian Hilbert spaces change with the regulator. A comparison construction must carry Ω_n to Ω strongly enough that vacuum orthogonality survives. Otherwise an apparent excited-state estimate can be distorted by changing the reference vacuum or by losing states in the comparison.

The source asks for nontrivial connected correlations, restoration of full four-dimensional Euclidean covariance, locality after reconstruction, clustering and independence from admissible auxiliary choices. The countable observable core is the place where these statements can be formulated precisely.

The exact finite vacuum is therefore a foundation for the programme, and also a constraint on every reduction used later. It must be transported consistently through projection, effective dynamics and continuum passage rather than removed informally when it becomes inconvenient.

Source: Auxiliary-time lattice gauge framework, §§27–30, printed pp.56–60.

Continue reading

• Exact Hamiltonians and their domains
• Why a sharp window divides the vacuum
• What neutrality does and does not remove
• Continuum reconstruction and gap transfer
• Reflection positivity from a vacuum Markov process
• A nonempty low space after vacuum puncture