## What is Quantum Field Theory?

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Download files for later. Send to friends and colleagues. Modify, remix, and reuse just remember to cite OCW as the source. Concepts and basic techniques are developed through applications in elementary particle physics, and condensed matter physics. This course is the first part of a three-course sequence. 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.

## Relativistic Quantum Field Theory I

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.

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.

Quantum Field Theory is the evolution of the so-called non-relativistic quantum mechanics of Schroedinger, Heisenberg, Planck, and the like. In order for it to work e. But the basis of the primacy of fields was well understood by physicists working in electromagnetism at the time. In this new interpretation, there was no such thing as a particle, but rather underlying field values. The study of these interactions is what is known as particle physics, and is the current most fundamental frontier in universally accepted quantum theories String theory, an extension of QFT from point-like quanta to line-like quanta, is likely the next frontier.

In this case, the same mathematical structure that in a calmer situation describes particle physics is hard to understand in the language of particles at all—the particle analogy has broken down. In the general case, the particle analogy is wildly nonsensical, so the fields must be the fundamental thing and indeed they are. The problem is aside from the particle analogy—the Feynman rules—we have in general no good way of working out the mathematics of Quantum Field Theory. Functionally, we can only calculate in the perturbative regime with the notable exception of Lattice QFT, which is extremely computationally intensive , so in the cases where we can actually compare the theory to experiment, the particle analogy makes perfect sense.

So in practical fact, QFT is a particle theory, not a field theory, since the way to make a calculation is to follow the particles. Quantum field theory QFT is essentially quantum mechanics for a system with an infinite number of degrees of freedom. The standard example is light inside a box with perfect mirrors on the six interior walls. A classical electromagnetic light wave can have a countably infinite number of possible wavelengths colors that fit perfectly inside the box along the three principal axes.

These add up to an infinite number of degrees of freedom. Each electromagnetic wave of a given color has a quantum state called a photon associated to it. This photon is the building block for a tower of states: But why mention Hindu gods and quantum fields? Photons can be destroyed or created in nature through the production of electron-positron pairs or through the annihilation of electron-positron pairs.

The operators that create and destroy photons within quantum field theory do so with discretion, however. Each mode direction and color of light in the box has a specific pair of creation and destruction operators associated to it, and they therefore constitute a field of operators over the infinite number of modes.

Among many other things, quantum field theory is distinguished from earlier quantum theories in that it allows one to model changes in the number of particles. It can model the creation and destruction of particles. Recall from classical quantum mechanics that the state of a particle is represented via a vector in a Hilbert space, i.

This is typically a space of complex-valued functions usually called wavefunctions.

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Physical observables like position and momentum take on an entirely different character. They cease to be simple properties that every particle possesses in all possible states. Instead, they become self-adjoint linear operators on the Hilbert space of state vectors, and an observable only acquires a definite value when the state happens to be an eigenvector of the associated operator in which case the value of the observable is typically the eigenvalue.

Moreover, one discovers that individual particles cannot really be distinguished from one another. One can only say that for a given quantum state a vector in the above space there exist zero or more particles of a given type in that state. The Pauli exclusion principle tells us that this count is always zero or one for fermions. For bosons it can in general be any nonnegative integer.

The collection of all this data constitutes a single state of the overall field, and it is seen that the possible states of the field comprise an infinite-dimensional state space often referred to as a Fock space. Note that there are a few other ways of constructing the state space of the field, but they often are considerably more complex than this technique and introduce issues with symmetrization or antisymmetrization of multi-particle states. This is a generic description of a procedure that is used in QFT to quantize all fields in physics except one, including but not limited to the electromagnetic field.

The one exception, of course, is the gravitational field. Moreover, particles that were not previously seen as excitations of a field are often viewed as such in QFT.

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Electrons, for instance, are viewed as excitations of an electron field, which admits its own creation and annihilation operators and interacts with exchanges energy with the electromagnetic field. This particle-oriented view of QFT leads to a brilliant picture of the interaction between electrons and the electromagnetic field, as promulgated by Feynman. Charged particles like electrons interact with the each other via the exchange of virtual photons. Virtual photons are messenger particles or force carriers that are passed between charged particles, ultimately conveying force.

This allows us to conceptualize the phenomena of electromagnetism entirely in terms of particles despite the theory being called field theory. For the electromagnetic field, it is true that the vacuum state indeed consists of no photons.

## Quantum Field Theory (Stanford Encyclopedia of Philosophy)

But for other fields, in which the particles can interact with each other, vacuum polarization occurs and makes the vacuum state much more difficult to describe. This page may be out of date. Save your draft before refreshing this page. Submit any pending changes before refreshing this page. Ask New Question Sign In. What is quantum field theory? Remember the wave-particle duality? Well, you might as well forget about it. In fact, there are no particles and no waves; just fields.

Both "particles" and "waves" are merely two ways in which we naively interpret quantum fields. There's one field for each type of particle. So one field for all photons in the universe, one field for all electrons, and so on. And these fields exist everywhere. To "extract" a particle from a field, you need to give the field energy.

If you give it enough energy, the field will go to a higher energy state. These states are what we interpret as particles. The point in the field where you gave it energy will look like a particle, and as the energy propagates through the field, it will look like the particle is moving. Some fields require more energy than others in order to create a particle.

The amount of energy is proportional to the mass of the associated particle. For example, a Higgs boson is much more massive than an electron. So electrons are very easy to create, but Higgs bosons are very hard to create. This is why it took us so long to discover the Higgs boson. We had to build a huge machine, the Large Hadron Collider , that was capable of giving the Higgs field enough energy to create Higgs bosons from it. Think of fields as candy machines. You put enough money in, you get a piece of candy. You put more money in, you get another candy.

The money in this analogy is energy, and the candies are particles. So if you have enough energy you can get as many particles that you want. Note that the money energy is always conserved: However, the number of candies particles is not conserved: There are 18 such machines that we know of. You can go, for example, to the electron machine, pay some money, and get an electron. And then you can go to the photon machine, pay some other amount of money, and get a photon. Crucially, the fields also interact and exchange energy with each other.

Think of an intricate network of tubes between the different machines, that can transfer coins from one machine to another. However, these exchanges all happen within the tubes, hidden from sight. No actual candies are exchanged in the process. The same thing happens with fields: Of course, it's actually much more complicated than that, but this is as far as the analogy can go Thank you for your feedback!