CPS 100 Lecture Notes - Lecture 6: Cartesian Coordinate System, Thermodynamic Limit, Photon
Quantum Field Theory
Quantum Field Theory (QFT) is the mathematical and conceptual framework for
contemporary elementary particle physics. In a rather informal sense QFT is the
extension of quantum mechanics (QM), dealing with particles, over to fields, i.e.
systems with an infinite number of degrees of freedom. (See the entry on quantum
mechanics.) In the last few years QFT has become a more widely discussed topic
in philosophy of science, with questions ranging from methodology and semantics
to ontology. QFT taken seriously in its metaphysical implications seems to give a
picture of the world which is at variance with central classical conceptions of
particles and fields, and even with some features of QM.
The following sketches how QFT describes fundamental physics and what the
status of QFT is among other theories of physics. Since there is a strong emphasis
on those aspects of the theory that are particularly important for interpretive
inquiries, it does not replace an introduction to QFT as such. One main group of
target readers are philosophers who want to get a first impression of some issues
that may be of interest for their own work, another target group are physicists who
are interested in a philosophical view upon QFT.
n contrast to many other physical theories there is no canonical definition of what
QFT is. Instead one can formulate a number of totally different explications, all of
which have their merits and limits. One reason for this diversity is the fact that QFT
has grown successively in a very complex way. Another reason is that the
interpretation of QFT is particularly obscure, so that even the spectrum of options
is not clear. Possibly the best and most comprehensive understanding of QFT is
gained by dwelling on its relation to other physical theories, foremost with respect
to QM, but also with respect to classical electrodynamics, Special Relativity Theory
(SRT) and Solid State Physics or more generally Statistical Physics. However, the
connection between QFT and these theories is also complex and cannot be neatly
described step by step.
If one thinks of QM as the modern theory of one particle (or, perhaps, a very few
particles), one can then think of QFT as an extension of QM for analysis of
systems with many particles—and therefore with a large number of degrees of
freedom. In this respect going from QM to QFT is not inevitable but rather
beneficial for pragmatic reasons. However, a general threshold is crossed when it
comes to fields, like the electromagnetic field, which are not merely difficult but
impossible to deal with in the frame of QM. Thus the transition from QM to QFT
allows treatment of both particles and fields within a uniform theoretical framework.
(As an aside, focusing on the number of particles, or degrees of freedom
respectively, explains why the famous renormalization group methods can be
applied in QFT as well as in Statistical Physics. The reason is simply that both
disciplines study systems with a large or an infinite number of degrees of freedom,
either because one deals with fields, as does QFT, or because one studies the
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thermodynamic limit, a very useful artifice in Statistical Physics.) Moreover, issues
regarding the number of particles under consideration yield yet another reason why
we need to extend QM. Neither QM nor its immediate relativistic extension with the
Klein-Gordon and Dirac equations can describe systems with a variable number of
particles. However, obviously this is essential for a theory that is supposed to
describe scattering processes, where particles of one kind are destroyed while
others are created.
One gets a very different kind of access to what QFT is when focusing on its
relation to QM and SRT. One can say that QFT results from the successful
reconciliation of QM and SRT. In order to understand the initial problem one has to
realize that QM is not only in a potential conflict with SRT, more exactly: the locality
postulate of SRT, because of the famous EPR correlations of entangled quantum
systems. There is also a manifest contradiction between QM and SRT on the level
of the dynamics. The Schrödinger equation, i.e. the fundamental law for the
temporal evolution of the quantum mechanical state function, cannot possibly obey
the relativistic requirement that all physical laws of nature be invariant under
Lorentz transformations. The Klein-Gordon and Dirac equations, resulting from the
search for relativistic analogues of the Schrödinger equation in the 1920s, do
respect the requirement of Lorentz invariance. Nevertheless, ultimately they are
not satisfactory because they do not permit a description of fields in a principled
quantum-mechanical way.
Fortunately, for various phenomena it is legitimate to neglect the postulates of
SRT, namely when the relevant velocities are small in relation to the speed of light
and when the kinetic energies of the particles are small compared to their mass
energies mc2. And this is the reason why non-relativistic QM, although it cannot be
the correct theory in the end, has its empirical successes. But it can never be the
appropriate framework for electromagnetic phenomena because electrodynamics,
which prominently encompasses a description of the behavior of light, is already
relativistically invariant and therefore incompatible with QM. Scattering experiments
are another context in which QM fails. Since the involved particles are often
accelerated almost up to the speed of light, relativistic effects can no longer be
neglected. For that reason scattering experiments can only be correctly grasped by
QFT.
Unfortunately, the catchy characterization of QFT as the successful merging of QM
and SRT has its limits. On the one hand, as already mentioned above, there also is
a relativistic QM, with the Klein-Gordon- and the Dirac-equation among their most
famous results. On the other hand, and this may come as a surprise, it is possible
to formulate a non-relativistic version of QFT (see Bain 2011). The nature of QFT
thus cannot simply be that it reconciles QM with the requirement of relativistic
invariance. Consequently, for a discriminating criterion it is more appropriate to say
that only QFT, and not QM, allows describing systems with an infinite number of
degrees of freedom, i.e. fields (and systems in the thermodynamic limit). According
to this line of reasoning, QM would be the modern (as opposed to classical) theory
of particles and QFT the modern theory of particles and fields. Unfortunately
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Document Summary
Quantum field theory (qft) is the mathematical and conceptual framework for contemporary elementary particle physics. In a rather informal sense qft is the extension of quantum mechanics (qm), dealing with particles, over to fields, i. e. systems with an infinite number of degrees of freedom. (see the entry on quantum mechanics. ) In the last few years qft has become a more widely discussed topic in philosophy of science, with questions ranging from methodology and semantics to ontology. Qft taken seriously in its metaphysical implications seems to give a picture of the world which is at variance with central classical conceptions of particles and fields, and even with some features of qm. The following sketches how qft describes fundamental physics and what the status of qft is among other theories of physics. Since there is a strong emphasis on those aspects of the theory that are particularly important for interpretive inquiries, it does not replace an introduction to qft as such.
