The Role of Field Operators in QFT

CHAPTER 7. FIELD INTERPRETATIONS OF QFT 98 mapping from each point of space to a field value.

As I indicated already the formal specificationφ(x, t) is not enough for something to be a field. Certain field equation need to be fulfilled. Without giving any further details I wish to point out just one extreme case why the formal specificationφ(x, t) cannot be sufficient. Considerφ(x, t) where φ(x, t) = 0∀x 6= ˜x with ˜x being a particular point in space. In this case φ(x, t) would just describe an ordinary point particle instead of a proper field.

One further information about fields should be supplied in order to make it understandable how one can come across the idea to think of fields as being the basic entities in the world. The intuitive notion of a field is that it is something transient and fundamentally different from matter.

However, in physics it is perfectly normal to ascribe energy and even mo-mentum to a pure field where no particles are present. This surprising feature shows how gradual the distinction between fields and matter can be.

CHAPTER 7. FIELD INTERPRETATIONS OF QFT 99 peculiarity of q numbers is the fact that they do not cummute in general, a fact whose details are condensed in the canonical commutation relations (CCRs). This peculiarity is in fact so characteristic that these relations are a sufficient information about the behaviour of a quantum particle.

Everything can be derived by specifying these relations.

In mathematical terminology the reason for the general non-commutation of q-numbers is that they are operators and not ordinary numbers. The order in which operators act on something does matter in general. In order to denote this difference operators get a hat. The transition from classical to quantum mechanics can thus be described as

x(t)→x(t)ˆ

and correspondingly for the momentum, where certain CCRs hold for their components.

Without going into the details let me just state that in a similar fash-ion the transitfash-ion from a classical field theory (like electromagnetism) to quantum field theory can be characterized by the transition

φ(x, t)→φ(x, t)ˆ

for the field and a corresponding transition for its conjugate field for both of which a certain specification of CCRs holds. In difference to a classical field φ(x, t) the basic fields ˆφ(x, t) of QFT are calledoperator-valued fields since to each point of space and time an operator is attached.

As one could see there is a formal analogy between classical and quan-tum fields. In both cases field values are attached to space-time points where these values are real-valued in the case of classical fields and operator-valued in the case of quantum fields. In technical terms the analogy reads as one between the mappings

x7→φ(x, t), x∈IR³ and

x7→φ(x, t),ˆ x∈IR³.

CHAPTER 7. FIELD INTERPRETATIONS OF QFT 100 This formal analogy between classical and quantum fields is one reason why QFT is taken to be a field theory. However, it has to be examined now whether this formal analogy actually justifies this conclusion.

In his paper “What the quantum field is not” Teller (1990) which be-came a central chapter of his later book An Interpretive Introduction to Quantum Field Theory Teller (1995) Teller puts considerable emphasis on a critique of this conclusion. He comes to the conclusion that ‘quantum fields’ lack an essential feature of all classical field theories so that the expression ‘quantum field’ is only justified on a “perverse reading” of the notion of a field. His reason for this conclusion is that in the case of quantum fields - in contrast to all classical fields - there are no definite physical values whatsoever assigned to space-time points. Instead, the assigned quantum field operators represent the whole spectrum of possi-ble values. They have, therefore, rather the status of observapossi-bles (Teller:

“determinables”) or general solutions. Something physical emerges only when the state of the system or when initial and boundary conditions are supplied.

I think Teller’s criticism of the standard gloss about operator-valued quantum fields has one justified and one unjustified aspect. The justified aspect is that quantum fields actually differ considerably from classical fields since the field values which are attached to space-time points have no direct physical significance in the case of the quantum field. However, and here I disagree with Teller, this fact is not due to the operator-valuedness of quantum fields as such. It was not to be expected anyway that one would only encounter definite values for physical quantities in QFT. QFT is, like QM, an inherently probabilistic theory, after all.

Nevertheless, even taking the probabilistic character of QFT into ac-count there still is the problem that we need quantum fields as well as state vectors in order to fix probabilistic properties. But I do not think that one can therefore conclude that quantum fields are not physically significant at all. It seems to me that physical significance of field quantities cannot be judged along classical distinctions. I think that there is more physical information encoded in quantum fields than Teller ascertains. But I agree

CHAPTER 7. FIELD INTERPRETATIONS OF QFT 101 with Teller that the field character of QFT is by no means as obvious as it first seems. The formal analogy between classical and quantum fields as such is not a fully convincing argument for a field interpretation of QFT.

If a field interpretation should actually yield the appropriate ontology for QFT than it seems that those objects which are called “quantum fields”

are not already the fundamental entities one is looking for, at least not alone.¹ And counting state vectors as basic entities as well spoils the idea of a field interpretation throughout since state vectors cannot be seen as fields in the sense of a mapping from space-time points to field values.