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The Interaction Hamiltonian

If interactions between particles are discrete, they will not be perfectly modelled by a continuous operator, but if the time scale for interactions is small, we might expect that time evolution will be modelled by a continuous operator on Fock space space, U(t) : F(t₀) → F(t₀ + t), to good approximation on observable timescales. In this case the arguments of section Time Evolution apply, Stone’s theorem can be used, and time evolution is given by

In a small time interval, Δt, there either is, or is not, an interaction. By the identification of the operations of vector space with weighted OR between uncertain possibilities, time evolution including the possibility of an interaction is described by a Hamiltonian, H, with
where H₀ is the free Hamiltonian, and H_int is a Hermitian operator describing an interaction between particles, called the interaction Hamiltonian.


In general, H_int will be a sum of terms for different types of interaction. For simplicity, I will consider only one type of interaction and spin indices will be suppressed. Spin is necessary for the full treatment of a specific interaction Hamiltonian, but has no bearing on the material on this page. Thus, the evolution of a state is given by

The Classical Correspondence

In the classical correspondence we study the behaviour of systems containing a large number, N, of quantum motions. Classical behaviour is the behaviour of a large population of quantum particles. It follows that classical properties are found from the expectations of the corresponding observables in the limit of large sample behaviour, as (not as sometimes stated; Planck’s constant is simply a change of scale from natural to conventional units and it would be meaningless to let it go to zero). For example, the centre of gravity of a macroscopic body is a weighted average of the positions of the elementary particles which constitute it. Schrödinger’s cat is definitely either alive or dead because, consisting as it does of a large number of elementary particles, its properties are expectations obeying classical laws. The state, or ket, simply encodes probability and the cat may be described as a superposition until the box is opened, but in the case of a classical cat this description makes no statement about reality and leads to no predictions different from standard probability theory.


Proof:  We have (from Stone’s theorem), for any ,
Since H is Hermitian,
Hence, by the definition of the Hermitian conjugate,
where the operators act to the left. Differentiate using the product rule,
which establishes Ehrenfest’s theorem. In particular, for an observable quantity with no explicit time dependence,

To give Ehrenfest’s theorem a relativistic form, we also need to describe the effect of change in position.


Proof:  Since space translation is the same for an observable operator, A(x), and the corresponding expectation, we have
and hence, differentiating from first principles,

Ehrenfest’s theorem describes change in the expectation of an observable, A. This assumes that measurement of the observable does not alter it’s evolution. In practice, measurements do alter evolution. Classical observables are continuously measurable (at least in principle). To fully describe the evolution of a classical observable, we will need to take into account the effect of measurement in addition to the evolution beween measurements described by the Hamiltonian.

The Interaction Picture

Without loss of generality, consider evolution from an initial state at time t₀ = 0. In order to study the effect of interactions it is convenient to separate interactions from free time evolution by applyinging the transformation
to states, and
to observables. This is the interaction picture In the absence of interaction, this would be the Heisenberg picture, and states would be constant in time. The interaction picture is useful for studying perturbations relative to to the behaviour of non-interacting particles, i.e. inertial matter, and implies that we are working in an inertial reference frame.

In the interaction picture, the interaction Hamiltonian is
The free Hamiltonian, H₀, is unchanged in the interaction picture, since an operator commutes with a function of itself.
So, the full Hamiltonian in the interaction picture is H₀ + H_I.

Observe that
So, time evolution is given in the interaction picture by the operator,
(It is usual to overload notation by using the same symbols in different pictures. It is important to keep track of which picture is being used). Differentiate,

The Hamiltonian Density

We assume that we can define a Hermitian interaction density operator, I(x), having the same effect on a matter anywhere and at any time, as required by the general principle of relativity. By the identification of addition with quantum logical OR, H_I(t) can be written as a sum (as usual on this site, an integral represents a finite sum with enough terms that taking more makes no practical difference to calculation).


The formal construction of the Hamiltonian density, I(x) is a major unresolved mathematical problem in quantum field theory. If particle interactions are discrete, then one might expect that a density might not exist. I will assume here that the formulae are valid in approximation for time intervals Δt much smaller than the resolution of measurement. I will consider the mathematical limit, Δt → 0, and implications of its possible non-existence later.

The Perturbation Expansion

We have the differential equation,
Integrate directly,
Substituting U iteratively back into the integral gives the Dyson expansion,

This can also be verified by differentiating. Each term is the derivative of the next multiplied by −iH(t). Substituting
gives
The integrals are strictly finite sums over the positions used for the basis of Hilbert space, and times .

Time Ordered Diagrams

Any operator on Fock space, F, can be written as a sum of products of creation and annihilation operators. The change of state associated with an interaction can be described as the annihilation of one state and the creation of another. Thus, a complete description of any process in interaction can be achieved through combinations of creation and annihilation operators. Expand the interaction density, I(x), as a sum of terms of the form
where and are creation and annihilation operators for the particles and antiparticles in the interaction.

i(x) can be represented diagrammatically as a vertex or node. The lines above the node correspond to creation operators, and those below the node correspond to annihilation operators.

The perturbation expansion for generates a braket between each annihilation operator, , and every earlier creation operator, , and every particle in , and a braket between every creation operator, , and every particle in the final state, , All other brakets are zero. These brakets can be represented graphically by connecting corresponding vertices. Lines representing particles are shown with arrows from bottom to top, and lines representing antiparticles with arrows from top to bottom. Then the n^th term of the perturbation expansion is a sum of terms, each represented as a time-ordered graph containing n vertices.

The perturbation expansion can be interpreted directly as a quantum logical statement, meaning that any number of interactions might be found taking place at any time and any position if we were to do a measurement. The sums (including the integral sums) simply represent OR between possibities. In this interpretation, I(x) describes the possibility that an interaction might be anywhere. It is not a quantised “matter field” which is, in some undefined sense, everywhere. Similarly, Feynman’s path integral, or “sum over all paths” has as natural interpretation that the sum over paths is a logical OR between the possible paths that might be detected if an experiment could be done to trace the path (not that a particle passes through all paths in spacetime, as described by Feynman in QED: The Strange Theory of Light and Matter). The time-ordered graphs give a pictorial representation of the same statement.

The Locality Condition

It can be seen that
Hence, we can write the perturbation expansion
The integrals are strictly finite sums over the positions used for the basis of Hilbert space, and times, . (Those very astute and well versed in analysis may see potential problems with this step in particular. For those more physically inclined, the justification usually given for the mathematics is that it leads to some very good physical predictions. I will not ignore that fact that it also leads to some serious inconsistencies which will have to be addressed ).


Under Lorentz transformation, the order of interactions, I(x_i), can be changed in the time-ordered product whenever x_i − x_j is space-like. Under the condition that the initial and final states are stable states of free particles, as in scattering experiments, the calculation of probabilities cannot be affected. The locality condition follows immediately.

In the interaction picture, Ehrenfest’s theorem states that the evolution of the expectation of an observable quantity which is not specifically time dependent (i.e. whose value depends only on the configuration of matter) is given by the commutator of the observable with the interaction Hamiltonian (repeat the proof in the interaction picture). Observable quantities are the result of physical measurement processes, which depend on the interactions of matter, so the commutator is zero outside the light cone. It follows that no observable effects may be transmitted faster than the speed of light. Of course, this argument is circular, since it depends on the premise of special relativity that information may not travel faster than the speed of light, but it serves to illustrate the physical meaning of the locality condition.

The locality condition also gives meaning to the statement that particles are point-like objects, since it shows that showing that two particles must meet at a point in order to interact. The meaning of the integral over space in the interaction Hamiltonian is logical disjunction, OR, meaning that a measurement to determine the position of the interaction could find any position, but it would always find the particles interacting at a point.

Conservation of Momentum

This is related to Noether’s theorem and, like it, depends on invariance under space translation. A separate proof is required because this is not a Lagrangian formulation and an action principle is not assumed.

For any plane wave states, ,
This reduces to a sum of terms containing a factor,
where π is a permutation of {1, …, m} and π' is a permutation of {m + 1, …, n}. This is
So, 3-momentum is conserved at each vertex. Between vertices, particles evolve according to the free Hamiltonian, H₀, which also conserves 3-momentum. So, momentum is always conserved.

It is straightforward to show that if the mass shell condition is obeyed, energy is not conserved at a vertex. It will be shown in the calculation of Feynman rules that energy is conserved for a system of interactions over large timescales.

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