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Linearity of Time Evolution

Hilbert space refers to measurement at time, t, so that , where a different Hilbert space is required at each time. Bold type will be used for 3-vectors. For the time being, it will be assumed that the scale on which quantum mechanics applies is such that curvature can be ignored, so that position coordinates can be denoted by displacement vectors in Minkowski spacetime. Position states at time x⁰ = t will be denoted .


If the state at time t₀ was either or , then it will evolve into either or at time t. Any weighting in quantum logical OR will be preserved, i.e., if
then
So,
In words, U is a linear operator.

Continuity of Time Evolution

Irrespective of whether a model of discrete particle interactions might appear continuous on the large scale, the evolution of kets is expected to be continuous because kets are not physical states of matter, but are rather probabilistic statements about what might happen in measurement, given current information. According to modern Bayesian ideas, probabilities describe our ideas concerning the likelihood of events. They are not a direct description of physical reality. Whether or not reality is fundamentally discrete, changes in probability can be properly described on a mathematical continuum. A discrete interaction will not lead to a discrete change in probability because we do not have information on when the interaction takes place. This being so, time evolution will be modelled by an continuous operator valued function of time, U. Together with the considerations below, continuity is sufficient to ensure differentiability (formal proof omitted).

At this point we have only states of a single, non-interacting particle. Subsequent pages will allow states containing an indefinite number of particles, and interactions betwen them. It is assumed that all phsysical systems will can be modeled in the same way. Since local laws of physics are always the same, and U does not depend on the state on which it acts, the evolution operator for a time interval t,
does not depend on t₀. We require that the evolution in an interval t₁ + t₂ is the same as the evolution in t₁ followed by the evolution in t₂, and is also equal to the evolution in t₂ followed by the evolution in t₁.
In a zero time interval, there is no evolution. So, U(0) does not change the state.
Using a negative value of t reverses time evolution (put t = t₁ = −t₂).

Unitarity of Time Evolution

Since states can be chosen to be normalised we may require that U conserves the norm, i.e. for all ,
Applying this to ,
By linearity of U,
By linearity of the inner product,
Similarly,
Combining these results,
So U is unitary.

Stone’s Theorem

The derivative of U is
This prompts the definition of the Hamiltonian operator, which does not depend on t.


We have
So
Since U is unitary, for a small time dt,
Ignoring terms in squares of dt, and using , ,
Using unitarity of U, we find that H is Hermitian, iH = H^†. We have the differential equation,
which has solution (as for a differential equation of a function)
This result was first proved by proved by Marshall Stone in 1932, and is known as Stone’s theorem.

The Heisenberg Picture

I have formulated quantum theory in such a way that states evolve in time and observable operators are assumed to be constant. This is the Schrödinger picture. A precisely equivalent formulation, the Heisenberg picture is found by a unitary transformation. States in the Heisenberg picture are defined by
and are constant. Then, an observable operator, A, in the Schrödinger picture, is given in the Heisenberg picture by
It is immediately clear that the result of calculation of probabilities is the same in both pictures,
Switching between the Schrödinger and Heisenberg pictures is simply a change of basis, and is precisely equivalent to coordinate transformation.

The Wave Function

It follows from linearity of U that the evolution of the ket from an initial state at time t₀ can be described in terms of its coefficients in a basis of position kets at time x⁰ = t, by using the resolution of unity,
On the assumption that the resolution of measurement may be arbitrarily fine, we define the wave function.



Proof: Differentiate the wave function using Stone’s theorem,

Newton’s First Law

According to the general principle of relativity, the laws of physics are the same for any two observers. In particular, any two observers can define coordinates in exactly the same way, using identical physical procedures. They formally describe probabilities for measurement results using quantum logic. Given the same information, they must calculate the same probabilities irrespective of their motion with respect to each other. So, the calculation of probabilities is constrained by relativistic considerations. In special relativity, it was found that 3-vectors must be replaced with 4-vectors. Applying this to plane wave states (assuming an inertial reference frame and Minkowski metric), we find that a plane wave evolves in the usual manner,
where E² = (p⁰)² = m² + p² for some constant m. Thus, p does not change in time,
establishing Newton’s first law. E is identified with energy and m with mass.

Wave Mechanics

To find the evolution of a ket from an initial state, , at t = 0, calculate the momentum space wave function using the resolution of unity,
Then, using the resolution of unity in momentum space,
By linearity of the time evolution operator, U,
The coefficients are
Thus, the momentum space wave function is constant in time. If we know the ket from a measurement at time t = 0, we can calculate the ket for a measurement at any other time, and hence probabilities for the results of measurement at any time, by the usual methods of wave mechanics.

This does not say that there is a physical wave. Quite the reverse, the appearance of complex numbers shows we are talking of mathematics, not of Nature, until such point as calculated probabilities are related to the freqencies of measurement results. It does show that quantum interference effects are simply the result of constructing a probability theory for measurements of position in such a way that any two observers, given the same information, will find the same probabilities for corresponding measurements. This suggests that quantum interference patterns are a manifestation of the fundamental structure of spacetime formed through the interactions of particles which make it possible to describe relative position. Exactly how this happens and leads also to curvature in general relativity and the force of gravity is the focus of study in relational quantum gravity.

Evolution of Quantum States ↑ The Dirac Equation →