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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. 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.

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 mc 2.

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. 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 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. 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 however, and this shall be the last turn, even this gloss is not untarnished.

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There is a widely discussed no-go theorem by Malament with the following proposed interpretation: Even the quantum mechanics of one single particle can only be consonant with the locality principle of special relativity theory in the framework of a field theory, such as QFT. Hence ultimately, the characterization of QFT, on the one hand, as the quantum physical description of systems with an infinite number of degrees of freedom, and on the other hand, as the only way of reconciling QM with special relativity theory, are intimately connected with one another.

The diagram depicts the relations between different theories, where Non-Relativistic Quantum Field Theory is not a historical theory but rather an ex post construction that is illuminating for conceptual purposes. Theoretically, [ i , ii , iii ], [ ii , i , iii ] and [ ii , iii , i ] are three possible ways to get from Classical Mechanics to Relativistic Quantum Field Theory. But note that this is meant as a conceptual decomposition; history didn't go all these steps separately. On the one hand, by good luck, so to say, classical electrodynamics is relativistically invariant already, so that its successful quantization leads directly to Relativistic Quantum Field Theory.

On the other hand, some would argue e. Malament that the only way to reconcile QM and SRT is in terms of a field theory, so that ii and iii would coincide. Note that the steps i , ii and iii , i. In other words, by these steps the nature of the physical entities the theories talk about may change fundamentally. Teller and Auyang are the first systematic monographs on the philosophy of QFT. The literature on the philosophy of QFT has increased significantly in the last decade. Besides a number of separate papers there are two new monographs, Cao and Kuhlmann , and one special issue May of Studies in History and Philosophy of Modern Physics.

Bain , Huggett and Ruetsche provide article length discussions on a number of issues in the philosophy of QFT. The crucial step towards quantum field theory is in some respects analogous to the corresponding quantization in quantum mechanics, namely by imposing commutation relations, which leads to operator valued quantum fields.

The starting point is the classical Lagrangian formulation of mechanics, which is a so-called analytical formulation as opposed to the standard version of Newtonian mechanics. Under these conditions the generalized momentum coincides with the usual mechanical momentum. In both cases, QM and QFT, requiring that the canonical variables satisfy certain commutation relations implies that the basic quantities become operator valued.

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From a physical point of view this shift implies a restriction of possible measurement values for physical quantities some but not all of which can have their values only in discrete steps now. It is not obvious that the equal-time commutation relations are Lorentz invariant but one can formulate a manifestly covariant form of the canonical commutation relations. If the field to be quantized is not a bosonic field, like the Klein-Gordon field or the electromagnetic field, but a fermionic field, like the Dirac field for electrons one has to use anticommutation relations.

While there are close analogies between quantization in QM and in QFT there are also important differences. Whereas the commutation relations in QM refer to a quantum object with three degrees of freedom, so that one has a set of 15 equations, the commutation relations in QFT do in fact comprise an infinite number of equations, namely for each of the infinitely many space-time 4-tuples x , t there is a new set of commutation relations.

This infinite number of degrees of freedom embodies the field character of QFT. In a certain sense the single particle wave functions have been transformed, via their reinterpretation as operator valued quantum fields, into observables. Afterwards the solutions to these single particle wave equations, which are states in relativistic QM, are considered as classical fields, which can be subjected to the canonical quantization procedure of QFT. In conclusion, it must be emphasized that both in QM and QFT states and observables are equally important.

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However, to some extent their roles are switched. While states in QM can have a concrete spatio-temporal meaning in terms of probabilities for position measurements, in QFT states are abstract entities and it is the quantum field operators that seem to allow for a spatio-temporal interpretation. See the section on the field interpretation of QFT for a critical discussion. Up to this point, the aim was to develop a free field theory. Doing so does not only neglect interaction with other particles fields , it is even unrealistic for one free particle because it interacts with the field that it generates itself.

For the description of interactions—such as scattering in particle colliders—we need certain extensions and modifications of the formalism. The immediate contact between scattering experiments and QFT is given by the scattering or S-matrix which contains all the relevant predictive information about, e. In order to calculate the S-matrix the interaction Hamiltonian is needed.

The Hamiltonian can in turn be derived from the Lagrangian density by means of a Legendre transformation. Using the interaction picture is advantageous because the equations of motion as well as, under certain conditions, the commutation relations are the same for interacting fields as for free fields. Therefore, various results that were established for free fields can still be used in the case of interacting fields. The central instrument for the description of interaction is again the S-matrix, which expresses the connection between in and out states by specifying the transition amplitudes.

Such probabilities can be checked in experiments. The canonical formalism of QFT as introduced in the previous section is only applicable in the case of free fields since the inclusion of interaction leads to infinities see the historical part. For this reason perturbation theory makes up a large part of most publications on QFT. The importance of perturbative methods is understandable realizing that they establish the immediate contact between theory and experiment. Although the techniques of perturbation theory have become ever more sophisticated it is somewhat disturbing that perturbative methods could not be avoided even in principle.

One reason for this unease is that perturbation theory is felt to be rather a matter of highly sophisticated craftsmanship than of understanding nature. Accordingly, the corpus of perturbative methods plays a small role in the philosophical investigations of QFT. What does matter, however, is in which sense the consideration of interaction effects the general framework of QFT.

An overview about perturbation theory is given in section 4. Some theories are distinguished by being gauge invariant , which means that gauge transformations of certain terms do not change any observable quantities. Requiring gauge invariance provides an elegant and systematic way of introducing terms for interacting fields. Moreover, gauge invariance plays an important role in selecting theories. The prime example of an intrinsically gauge invariant theory is electrodynamics.

The important point in the present context is that given the identification 3.

In order to see that, consider the so-called gauge transformations. Inserting the transformed potential s into equation s 3. Since only the electric field E and the magnetic field B , and quantities constructed from them, are observable, whereas the vector potential itself is not, nothing physical seems to be changed by a gauge transformation because it leaves E and B unaltered. Note that gauge invariance is a kind of symmetry that does not come about by space-time transformations. Equations 3. It turned out that requiring invariance under local gauge transformations supplies a systematic way for finding the equations describing fundamental interactions.

For instance, starting with the Lagrangian for a free electron, the requirement of local gauge invariance can only be fulfilled by introducing additional terms, namely those for the electromagnetic field. This is an important basis for unification programs, as is the analogy to general relativity where a local gauge symmetry is associated with the gravitational field. Moreover, it turned out that only gauge invariant quantum field theories are renormalizable.

All this can be taken to show that a mathematically rich theory, with surplus structures, can be very valuable in the construction of theories. Auyang emphasizes the general conceptual significance of invariance principles; Redhead and Martin focus specifically on gauge symmetries. Healey and Lyre and discuss the ontological significance of gauge theories, among other things concerning the Aharanov-Bohm effect and ontic structural realism. In the s a program emerged in which the theories of the standard model of elementary particle physics are considered as effective field theories EFTs which have a common quantum field theoretical framework.

EFTs describe relevant phenomena only in a certain domain since the Lagrangian contains only those terms that describe particles which are relevant for the respective range of energy. EFTs are inherently approximative and change with the range of energy considered.

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EFTs are only applicable on a certain energy scale, i. Influences from higher energy processes contribute to average values but they cannot be described in detail. This procedure has no severe consequences since the details of low-energy theories are largely decoupled from higher energy processes. Both domains are only connected by altered coupling constants and the renormalization group describes how the coupling constants depend on the energy. The main idea of EFTs is that theories, i. The physics changes by switching to a different energy scale, e.

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The dependence of theories on the energy scale distinguishes QFT from, e. Nevertheless, laws from different energy scales are not completely independent of each other. A central aspect of considerations about this dependence are the consequences of higher energy processes on the low-energy scale.

On this background a new attitude towards renormalization developed in the s, which revitalizes earlier ideas that divergences result from neglecting unknown processes of higher energies. Low-energy behavior is thus affected by higher energy processes. Since higher energies correspond to smaller distances this dependence is to be expected from an atomistic point of view. According to the reductionist program the dynamics of constituents on the microlevel should determine processes on the macrolevel, i.

However, as, for instance hydrodynamics shows, in practice theories from different levels are not quite as closely connected because a law which is applicable on the macrolevel can be largely independent of microlevel details. For this reason analogies with statistical mechanics play an important role in the discussion about EFTs.

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The basic idea of this new story about renormalization is that the influences of higher energy processes are localizable in a few structural properties which can be captured by an adjustment of parameters.