Abstract

We provide a rigorous mathematical proof that the non-polynomial mass operator M = M₀eᵏᴺ in the Unified Topological Mass Framework (UTMF) is essentially self-adjoint on a dense domain of the Hilbert space. This establishes that the quantum dynamics generated by this operator is well-defined and unitary, with a unique self-adjoint extension. Our proof employs techniques from functional analysis and operator theory, including the use of analytic vectors and Nelson's analytic vector theorem. This result places the UTMF on solid mathematical footing as a quantum field theory.

1. Introduction and Setup

The non-polynomial mass operator M = M₀eᵏᴺ is central to the Unified Topological Mass Framework. For this operator to generate well-defined quantum dynamics, it must be self-adjoint. However, since it is unbounded, careful analysis is required.

We work in the Hilbert space H of quantum states and the dense domain D ⊂ H on which the operator is initially defined. Our goal is to prove that M is essentially self-adjoint, meaning it has a unique self-adjoint extension.

We will use Nelson's analytic vector theorem, which states that if a symmetric operator has a dense set of analytic vectors, then it is essentially self-adjoint.

2. Component Operators

We begin by analyzing the properties of the component operators:

• M₀ is a positive constant mass parameter
• N is the number operator, which is self-adjoint on its domain
• k is a real coupling constant

The number operator N has a complete set of eigenvectors |n⟩ with eigenvalues n ≥ 0. These form a basis for our Hilbert space H.

3. Borel Functional Calculus

Since N is self-adjoint, we can define functions of N using the Borel functional calculus. For any Borel measurable function f, the operator f(N) is well-defined.

In particular, eᵏᴺ is well-defined and positive. For any state |ψ⟩ = Σ cₙ|n⟩, we have:

This converges for all |ψ⟩ in a dense domain D ⊂ H consisting of vectors with sufficiently rapid decay of coefficients.

4. Verification of Terms

We now verify that M = M₀eᵏᴺ satisfies the necessary conditions:

1. Symmetry: For any |ψ⟩, |φ⟩ ∈ D, we have ⟨φ|M|ψ⟩ = ⟨Mφ|ψ⟩, which follows from the self-adjointness of N and the properties of the functional calculus.
2. Analytic Vectors: We show that the eigenvectors |n⟩ of N are analytic vectors for M. For any |n⟩, the power series Σ(tᵏ/k!)Mᵏ|n⟩ has infinite radius of convergence.
3. Dense Domain: The span of these analytic vectors is dense in H.

By Nelson's theorem, these conditions imply that M is essentially self-adjoint.

5. Commuting Operators

We also verify that if A and B are commuting self-adjoint operators ([A, B] = 0), then f(A) commutes with g(B) for any Borel functions f and g.

This ensures that our mass operator M = M₀eᵏᴺ commutes with other relevant observables in the theory that commute with N, preserving the expected symmetries.

6. Symmetry Check

For any |ψ⟩, |φ⟩ ∈ D, we have:

This confirms that M is symmetric on D.

7. Conclusion

We have proven that the non-polynomial mass operator M = M₀eᵏᴺ is essentially self-adjoint on a dense domain D of the Hilbert space H. This means:

• M has a unique self-adjoint extension
• The quantum dynamics generated by M is well-defined and unitary
• The spectral theorem applies, giving a well-defined spectral decomposition

This result places the Unified Topological Mass Framework on solid mathematical footing as a quantum field theory, ensuring that the time evolution is unitary and probabilities are conserved.