Local quantum fields

For the attempt to take over the reasoning from the last section into QFT, immedi- ately several problems arise. First, fields in QFT are typically defined over all of spacetime and have a value different from zero everywhere. In practice often cutoffs are introduced, viz., fields are only evaluated in a certain spacetime region, and it is assumed that the fields fall off quickly towards infinity. Nevertheless, if again the non-zero part of the field is regarded as constituting the world line, it follows that world lines will typically extend over all of spacetime. Even worse, if there is more than one field, then the corresponding world lines will constantly intersect and it is not clear how the unintuitive conclusion could be avoided that everything interacts causally with everything at all places and times.

The second problem is that it is not even clear whether it makes sense to interpret quantum field values as world lines. In the field formulation of Newtonian mechanics from the last section, a positive field value can be interpreted as the field being ‘impenetrable’ at that point, which means that there is a particle (cf. Stein 1970, p.

277). Force fields in classical physics assign a field strength to every spacetime point, which gives the force that a test particle would feel at that point. To the contrary, fields in QFT only assign operators to every spacetime point, for which an equally

direct interpretation seems to be impossible. This situation is made worse by the fact that there is no position operator in QFT, which could be interpreted as giving the probability for there being an object at point x in space (cf. Ticciati 1999, ch. 1.6). This makes Dieks (2001) doubt whether the manifold, on which the quantum field is defined can rightly be called spacetime at all. Another reaction is that of Teller (1995) who tries to give operators at each spacetime point a direct physical interpretation as determinable properties. However, it can be doubted whether Teller’s approach is successful (cf. Wayne 2002; Fleming 2002). As a result, even if the first problem can be answered and world lines be defined with the help of the field, it still remains unclear whether the world lines can be regarded as belonging to physical objects or not.

The most immediate, but naive, answer to the first problem might be to exploit whether fields can be defined in such a way that they only have a non-zero value within a spacetime region O and zero in the complement region O0_{, that is the so-}

called Newton-Wigner localisation. This solution however fails, since it is in conflict with special relativity, as Malament (1996) shows.1 _{This problem is evaded by a}

proposal due to Knight (1961).2 _{He first points out that all measurable quantities in}

QFT are the results of products of fields acting on a certain state: ha| ˆ(x1), . . . , ˆ(xn)|ai .

Based thereon, Knight defines the state |ai of a field ˆ(x) as strictly localised in a region O, if outside of that region, in O0_{, it is the vacuum state |0i, that is:}

ha| ˆ(xi)|ai = h0| ˆ(xi)|0i for all xi 2 O0 .

Here a slight ambiguity arises as to whether we say that either the state or the field is localised. But nothing depends on it, since from the definition it is clear what localisation means and mathematically it can be constructed either by a suitable manipulation of the state or the field.

However, as Haag & Swieca (1965) point out, the definition of Knight alone is not tenable, because of the Reeh-Schlieder theorem. The latter exploits the entanglement of the vacuum state and has as a consequence that “we can approximate any state vector by linear combinations of vectors describing states ‘strictly localized’ in some region O.” (Haag 1996, p. 254) Therefore, states that satisfy Knight’s requirement will in general not be localised in O. On the other hand, the correlations of the vacuum state fall off very quickly with distance and to use them to create states over long distances requires “prohibitively high-energy states” (Wallace 2001, p. 10)

1_{Fleming & Butterfield (1999) have responded to Malament that Newton-Wigner localisation is}

possible, if localisation is always defined with respect to a spacelike hyperplane. This approach, however, has been criticised by Halvorson (2001).

2_{It should be noted that the definitions of Knight (1961) and, later to be discussed, Haag (1996)}

and Wallace (2001) are strictly only shown to be valid for massive bosonic scalar fields, but it can be expected that a generalisation to other fields is in principle unproblematic (cf. Knight 1961, p. 460 and Wallace 2001, p. 41).

This motivates that we weaken strict localisation to what Haag & Swieca (1965) call ‘essential localisation’ and Wallace (2001) calls ‘effective localisation’. The idea is not to require that a state is the vacuum state outside of its region of localisation, but only that it approximates the vacuum state. If the probability for a state to be localised in a sphere around region O falls off sufficiently fast, one can choose a certain distance from O after which the probability is extremely small. Wallace (2001, p. 10) puts these considerations together to the following definition:

Effective localisation (qualitative form): A state |ai is effectively localised in a spatial region ⌃i iff for any function ˆf of field operators ˆ, ˆ⇡, ha| ˆf|ai h0| ˆf|0i

is negligibly small when ˆf is evaluated for field operators outside ⌃i, compared

to its values when evaluated for field operators within ⌃i. [notation changed]

Now, the definition after which point exactly the probability for localisation outside of ⌃i is negligible is to some extent arbitrary, and I will say more on the justification

for this form of localisation in the next section. For the moment it is sufficient that, following Wallace (2001, p. 27 f.), from the rate that the entanglement of the vacuum state decreases, we can expect the diameter of ⌃i to be on a length scale of the order

of the Compton wavelength. As also stressed by Wallace (2001, p. 30), an important property of effectively localised states is that they remain at least approximately localised under time translations.

Coming now back to the initial problem of how the notion of world lines might be explicated in QFT, it is fairly obvious how effective localisation helps: A world line is the spacetime region O in which a field ˆ is effectively localised. However, one might object that this definition of world lines is not very enlightening, since operator valued fields are no physical entities and therefore cannot have a world line in spacetime. This is the second problem, stated at the beginning.

This concern is certainly well justified, but that a quantum field assigns operators to each spacetime point is not all what one can say about that field. In general, “[a]ll quantum field theories [...] model localization by making observables dependent on position in spacetime” (Halvorson & Clifton 2002, 18). Therefore any observable ˆO(x), defined at spacetime point x, with a non-vanishing expectation value is sufficient to refer to an entity localised at x.3 _{Another virtue of the quantum field is that}

it specifies the energy-momentum of a state. This usually is calculated by letting the Hamilton operator, which is a product of creation and annihilation operators, act on the vacuum state. For an effectively localised state it trivially follows that (nearly) the whole energy-momentum is contained in the region where the state is localised. Now this will not ease someone who is concerned about ontology, like, e.g., Teller, because it does not tell us what it is that is localised. Nevertheless, since my main focus lies on causation, knowing that something is localised, whatever it is, is a major step forward.

In conclusion, even in QFT it is possible to define localised states with a localised energy-momentum. Furthermore, given that the CQT describes causation with the

help of world lines and the exchange of energy-momentum, the fact that we can define world lines that have an associated energy-momentum in QFT could serve as the starting point for some kind of transference theory in the context of QFT. Having said that, this discussion only treated free fields, and the remaining task is now to take a closer look on interactions, but before doing that, another serious worry needs to be discussed.

I take it that Wallace has chosen the adjective ‘effective’ for his version of localisa- tion with care, since it has some pragmatist connotations. Indeed, Wallace’s main concern is to explain the emergence of small particles through localised states. He does not claim that what there is are particles. At least as presented here, effective localisation has an ad hoc character and one might well question what justifies the claim that a state is localised in a region, even though the probability to find the state outside that region, though very small, is non-zero. As already mentioned, the choice of the region’s size in which the state is said to be localised is prima facie arbitrary. A possible answer to this question is what I will engage with in the next section.