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The Vacuum State

So far the language of quantum logic has only been used to describe the result of hypothetical measurement of single particles. We also want to be able to describe situations with any number of particles, and the situation where there is no particle, namely the vacuum.


is the empty ket and is informally known as the vacuum state. Since this is a language to describe hypothetical measurement, we may assume that the results of both measurements are meaningful and accurate. The structure of quantum logic for single particles has already implicitly assumed that the hypothetical first and second measurements are measurements of the same particle. In real experiments this is not always the case. The separation of “noise” from genuine results is a problem which gives experimentalists no end of trouble, but it is not something which need concern us here. We want theory to describe only genuine results. The practical problems of comparing theory to experiment can safely be left to the experimentalist. Thus, we assume “perfect” measurement in which the finding of no particle means there actually is no particle, not that there might have been a particle but we failed to detect it. If the first measurement were to yield the vacuum state, so would the second.

Let H⁰ be the space spanned by . H⁰ is a one dimensional vector space. Because multiplication by scalars only has meaning in association with the weighting in OR, there is no difference in meaning between member clauses, , of H⁰.

Many Particles

To describe states of many particles, the hypothetical measurements must be able to detect all the particles in a system. In this case the results are described using grammatical or logical conjunction, AND. A natural structure for clauses ANDing the results of measurement for several particles is provided by the tensor product. For clarity I will suppress spin indices. Thus the tensor, , means “if measured position of the first particle were x and measured position of the second particle were y…”.



In practice we may not know how many particles there are, and the number of particles may be subject to change. To accommodate this within the structure of vector space we form a direct sum. A direct sum means that we take a set of all the basis elements of all the spaces, Hⁿ, and use it as a basis for a new vector space by introducing new operations of addition and multiplication space, while maintaining compatibility with existing addition and multiplication by scalars. In terms of the formal language this means we can make statements about an uncertain number of particles, using weighted logical OR, “If, for each of n or m particles, but more likely n than m, …”, etc.


As usual the sum means to a large value of n, greater than any number of particles we may actually wish to consider.

Independence of Particles in Measurement

Since an n particle state cannot be an m particle state, the braket between states of different numbers of particles is zero. For two n-particle states, and , the braket is given by the standard formula for the inner product of a tensor space,
Thus, the probability of the result of actual measurements on n particles factorises into n probabilities for the results of measurements on the particles individually,
In probability theory this is the condition for statistical independence, and means that, in the absence of other constraints, particles behave independently of each other in measurement.

Identical Particles

For identical particles, there is no way of telling which is which in a measurement. Switching identical particles makes no difference to the physical situation. We extend the language to cover this situation.


For a two particle system , and we require that and are both TRUE,
So, κ² = 1, and κ = ±1. The value of κ is said to determine the statistics of the type of particle under study. κ = 1 determines Bose-Einstein statistics, and κ = −1 determines Fermi-Dirac statistics.


Parastatistics have been considered for particles appearing in states of more than two particles. They may have applicability for quarks, but go beyond anything considered here.

Since position states are a basis for H²,
a and b are chosen so that each result is equally likely, i.e |a| = |b|. An overall scale factor has no meaning and for a normalised state, , we find, for Bosons,
and for Fermions,

The Pauli Exclusion Principle

The sum of symmetric states is again a symmetric state, and likewise the sum of antisymmetric states is an antisymmetric state. Multiplication by a scalar also does not affect symmetry. So, for any states and in H¹, we have
It follows at once that for Fermions.
This is summarised in the Pauli exclusion principle.


The Pauli exclusion principle is the major principle responsible for the structures of matter in our environment. It is the reason for the structure of electron shells around the nucleus of the atom, the reason solid objects cannot occupy the same position in space, and the reason why gas particles rebound of each other and generate the gas laws. It also governs the behaviour of stars when thermonuclear combustion ceases, leading to white dwarfs supported by the exclusion principle between electrons (electron degeneracy pressure), and to neutron stars supported by the exclusion principle between neutrons (neutron degeneracy pressure).

Creation and Annihilation Operators

In interactions, particles may be created, as described by creation operators, and particles may be destroyed. The destruction of a particle in interaction is described by the action of an annihilation operator. A change of state of a particle 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. Creation and annihilation operators are linear operators, and incorporate the idea that when a particle is created it is impossible to distinguish it from any existing particle of the same type, so that they automatically (anti)symmetrise states of identical particles. A creation operator is closely associated with the state which it creates, and will be denoted as a ket, with an underline to distinguish it from a state. I have found that this use of ket notation simplifies and clarifies many formulae in qed. By linearity, it is sufficient to define its action on basis states.




When a particle is created in a negative energy state, it goes backwards in time and the annihilation of an antiparticle is seen. Similarly the annihilation of a negative energy state is seen as the creation of an positive energy antiparticle. Creation and annihilation operators for antiparticles will play opposite roles to those for particles, and are designated with an overline.


States of Many Particles ↑ Particle Interactions →