[Paper Review] Time in Quantum Mechanics

C. A. Moyer|arXiv (Cornell University)|May 23, 2013

Quantum Mechanics and Applications5 references3 citations

TL;DR

This paper proposes a novel quantum mechanical framework where time is treated as a physical observable through a time basis of states, called 'time states' or 'quantum histories,' generated by the Hamiltonian as the time-translation operator. Despite Pauli's theorem forbidding self-adjoint time operators for semi-bounded Hamiltonians, the paper shows that time states form a complete, unitary basis for any system, enabling a statistical description of event times via a POVM. The key contribution is establishing quantum histories as a legitimate alternative to position and momentum bases, even when time operators cannot be consistently defined.

ABSTRACT

The failure of conventional quantum theory to recognize time as an observable and to admit time operators is addressed. Instead of focusing on the existence of a time operator for a given Hamiltonian, we emphasize the role of the Hamiltonian as the generator of translations in time to construct time states. Taken together, these states constitute what we call a timeline, or quantum history, that is adequate for the representation of any physical state of the system. Such timelines appear to exist even for the semi-bounded and discrete Hamiltonian systems ruled out by Pauli's theorem. However, the step from a timeline to a valid time operator requires additional assumptions that are not always met. Still, this approach illuminates the crucial issue surrounding the construction of time operators, and establishes quantum histories as legitimate alternatives to the familiar coordinate and momentum bases of standard quantum theory.

Motivation & Objective

• To address the foundational problem of time in quantum mechanics, where time is typically a parameter, not an observable.
• To challenge the conventional view that time cannot be an observable due to Pauli's theorem.
• To construct a complete, unitary basis of time states for any quantum system, even those with semi-bounded or discrete Hamiltonians.
• To show that time states form a valid quantum history basis, enabling statistical prediction of event times.
• To clarify the distinction between a time basis and a time operator, and to identify conditions under which the latter can exist.

Proposed method

• Introduces time states |τ⟩ labeled by real system time τ, defined by the requirement that the Hamiltonian generates translations: |τ+α⟩ = exp(−iĤα)|τ⟩.
• Establishes the time basis as complete via the resolution of identity: ∫|τ⟩⟨τ|dτ = 1.
• Derives covariance of the time basis: ⟨τ|ψ(t)⟩ = ⟨τ−t|ψ(0)⟩, ensuring time-translation invariance.
• Uses the time basis to define a POVM for event time measurements, avoiding the need for a self-adjoint time operator.
• Analyzes the conditions under which a self-adjoint time operator can exist, showing they are not always satisfied.
• Applies the formalism to free particles in 3D, deriving a canonical time operator and verifying [T̂, Ĥ] = i.

Experimental results

Research questions

• RQ1 Can a complete, unitary basis of time states be constructed for any quantum system, even when Pauli’s theorem forbids a self-adjoint time operator?
• RQ2 How can event time distributions be consistently described in quantum mechanics without assuming a time operator?
• RQ3 What is the relationship between the time basis and the existence of a self-adjoint time operator?
• RQ4 Can a time operator be constructed for the free particle in three dimensions, and does it satisfy the canonical commutation relation with the Hamiltonian?
• RQ5 Does the time basis provide a physically meaningful alternative to the standard position and momentum bases in quantum theory?

Key findings

• A time basis of states |τ⟩ exists for all quantum systems, including those with semi-bounded or discrete Hamiltonians, bypassing Pauli’s theorem.
• The time basis satisfies the resolution of identity and covariance, ensuring consistent statistical predictions for event times.
• The time basis is orthogonal in a weak sense and allows the representation of any physical state via ∫⟨φ|τ⟩⟨τ|ψ⟩dτ.
• For the free particle in three dimensions, a time operator T̂_3d-free is constructed and shown to satisfy the canonical commutation relation [T̂, Ĥ] = i.
• The time basis provides a valid alternative to position and momentum bases, even when a self-adjoint time operator does not exist.
• The existence of a time basis does not imply the existence of a time operator; additional physical and mathematical conditions are required.