Assuming that one can determine the (unique) quantum numbers of the vacuum, Quantum Field Theory then uses field operators which operate on the vacuum state to create field quanta. The resultant state can be further operated upon to both create and destroy (additional) field quanta. These operators are associated with complex numerical values and do not "commute": the result of applying two operators to the same state in different orders will not necessarily be the same.
In fact, the complex values associated with the operators are "distributions". These values must be averaged out over a finite spacetime region in order to be finite and hence meaningful.
The fact that the numerical values associated with the operators (and hence the field quanta) are complex implies that they have two conjugate degrees of freedom. It is common to use a coordinate/momentum conjugate representation in quantum field theory. In such a representation, the real part (ie., the amplitude) of the operator value is typically a function of momentum while the imaginary part (ie., the phase) is then a function of both the coordinate and momentum. Another common representation is in terms of energy modes, where the phase is a function of energy and time (the conjugate variables in this case). One mathematical result of this use of conjugate variables is the "uncertainty relation": the range of values of one variable required to describe the object is inversely proportional to the range of values required of its conjugate. Hence if a small range of position values is used to describe a particle, a large range of momentum values is required as well.
For sufficiently complicated collections of interacting quanta, the phase relationships (except in extrordinary "macroscopic quantum systems") vary rapidly, with the result that the overall collection possesses only a single degree of freedom, the amplitude (the phase of the collection is essentially random, and averages to zero).
Quantum Gravity Concept Map Index:
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©1997, Kenneth R. Koehler. All Rights Reserved.