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Quantum field theory

Quantum field theory (QFT) is the application of quantum mechanics to fields. It provides a theoretical framework, widely used in particle physics and condensed matter physics, in which to formulate consistent quantum theories of many-particle systems, especially in situations where particles may be created and destroyed. Non-relativistic quantum field theories are needed in condensed matter physics— for example in the BCS theory of superconductivity. Relativistic quantum field theories are indispensable in particle physics (see the standard model), although they are known to arise as effective field theories in condensed matter physics.

Origin

Quantum field theory originated in the problem of computing the power radiated by an atom when it dropped from one quantum state to another of lower energy. This problem was first examined by Max Born and Pascual Jordan in 1925. In 1926, Max Born, Werner Heisenberg and Pascual Jordan wrote down the quantum theory of the electromagnetic field neglecting polarization and sources to obtain what would today be called a free field theory. In order to quantize this theory, they used the canonical quantization procedure. In 1927, Paul Dirac gave the first consistent treatment of this problem. Quantum field theory followed unavoidably from a quantum treatment of the only known classical field, ie, electromagnetism. The theory was required by the need to treat a situation where the number of particles changes. Here, one atom in the initial state becomes an atom and a photon in the final state.

It was obvious from the beginning that the quantum treatment of the electromagnetic field required a proper treatment of relativity. Jordan and Wolfgang Pauli showed in 1928 that commutators of the field were actually Lorentz invariant. By 1933, Niels Bohr and Leon Rosenfeld had related these commutation relations to a limitation on the ability to measure fields at space-like separation. The development of the Dirac equation and the hole theory drove quantum field theory to explain these using the ideas of causality in relativity, work that was completed by Wendell Furry and Robert Oppenheimer using methods developed for this purpose by Vladimir Fock. This need to put together relativity and quantum mechanics was a second motivation which drove the development of quantum field theory. This thread was crucial to the eventual development of particle physics and the modern (partially) unified theory of forces called the standard model.

In 1927 Jordan tried to extend the canonical quantization of fields to the wave function which appeared in the quantum mechanics of particles, giving rise to the equivalent name second quantization for this procedure. In 1928 Jordan and Eugene Wigner found that the Pauli exclusion principle demanded that the electron field be expanded using anti-commuting creation and annihilation operators. This was the third thread in the development of quantum field theory— the need to handle the statistics of multi-particle systems consistently and with ease. This thread of development was incorportated into many-body theory, and strongly influenced condensed matter physics and nuclear physics.

What QFT is

Just as quantum mechanics deals with operators acting upon a (separable) Hilbert space, QFT also deals with operators acting upon a Hilbert space. However, in the case of QFT, the operators are generated by what is known as operator-valued fields, that is, operators which are parametrized by a spacetime point. Intuitively, this means that operators can be localized. This definition applies even to the cases of theories which aren't quantizations, and as such, is pretty general.

This is sometimes stated as "position is an operator in QM but is a parameter in QFT" but this statement, while accurate, can be very misleading. QM deals with particles and one of the properties of a particle is its position as a function of time and in QM, this becomes the position operator as a function of time (it's constant in the Schrödinger picture and varying in the Heisenberg picture). QFT, on the other hand, deals with fields on a fundamental level and particles only emerge as localized excitations (aka quanta aka quasiparticles) of the ground state (aka the vacuum) and it's precisely these quantum fields which correspond to the operator valued functions. Put more simply, instead of looking at the operators generated by

and ,

we now look at operators generated by

And just as in QM, we may work in the Schrödinger picture, the Heisenberg picture or the interaction picture (in the context of perturbation theory). Only the Heisenberg picture is manifestly Lorentz covariant.

The energy is given by the Hamiltonian operator, which can be generated from the quantum fields, and corresponds to the generator of infinitesimal time translations. (the condition that the generator of infinitesimal time translations can be generated by the quantum fields rules out many unphysical theories, which is a good thing) We further assume that this Hamiltonian is bounded from below and has a lowest energy eigenstate (this rules out theories which are unstable and have no stable solutions, which is also a good thing), which may or may not be degenerate. (although there are physical QFTs which have a lower bound to the Hamiltonian but don't have a lowest energy eigenstate, like N=1 super QCD theories with too few quarks...) This lowest energy eigenstate is called the vacuum in particle physics and the ground state in condensed matter physics. (QFT appears in the continuum limit of condensed matter systems)

This simple explanation of what QFT really is, is often obscured in treatments which jump straight to the path integral approach, which is a good computational technique but often obscures the underlying ideas.

QFT most definitely isn't the same thing as classical field theory or classical field theory with some "minor" quantum corrections, which is a mistake many high energy physicists are prone to making at times, especially when working in the semiclassical approximation.

Technical statement

Quantum field theory corrects several limitations of ordinary quantum mechanics, which we will briefly discuss now. The Schrödinger equation, in its most commonly encountered form, is

where denotes the quantum state (notation) of a particle with mass m, in the presence of a potential V.

The first problem occurs when we seek to extend the equation to large numbers of particles. As described in the article on identical particles, quantum mechanical particles of the same species are indistinguishable, in the sense that the state of the entire system must be symmetric (bosons) or antisymmetric (fermions) when the coordinates of its constituent particles are exchanged. These multi-particle states are extremely complicated to write. For example, the general quantum state of a system of N bosons is written as

where are the single-particle states, Nj is the number of particles occupying state j, and the sum is taken over all possible permutations p acting on N elements. In general, this is a sum of N! (N factorial) distinct terms, which quickly becomes unmanageable as N increases. Large numbers of particles are needed in condensed matter physics where typically the number of particles is on the order of Avogadro's number, approximately 1023.

The second problem arises when trying to reconcile the Schrödinger equation with special relativity. It is possible to modify the Schrödinger equation to include the rest energy of a particle, resulting in the Klein-Gordon equation or the Dirac equation. However, these equations have many unsatisfactory qualities; for instance, they possess energy eigenvalues which extend to –∞, so that there seems to be no easy definition of a ground state. Such inconsistencies occur because these equations neglect the possibility of dynamically creating or destroying particles, which is a crucial aspect of relativity. Einstein's famous mass-energy relation predicts that sufficiently massive particles can decay into several lighter particles, and sufficiently energetic particles can combine to form massive particles. For example, an electron and a positron can annihilate each other to create photons. Such processes must be accounted for in a truly relativistic quantum theory. This problem brings to the fore the notion that a consistent relativistic quantum theory, even of a single particle, must be a many particle theory.

Quantizing a classical field theory

Canonical quantization

Quantum field theory solves these problems by consistently quantizing a field. By interpreting the physical observables of the field appropriately, one can create a (rather successful) theory of many particles. Here is how it is:

1. Each normal mode oscillation of the field is interpreted as a particle with frequency f.

2. The quantum number n of each normal mode (which can be thought of as a harmonic oscillator) is interpreted as the number of particles.

The energy associated with the particle is therefore = . With some thought, one may similarly associate momenta and position of particles with observables of the field.

Having cleared up the correspondence between fields and particles (which is different from non-relativistic QM), we can proceed to define how a quantum field behaves.

Two caveats should be made before proceeding further:

Each of these "particles" obeys the usual uncertainty principle of quantum mechanics. The "field" is an operator defined at each point of spacetime.

Quantum field theory is not a wildly new theory. Classical field theory is the same as classical mechanics of an infinite number of dynamical quantities (say, tiny elements of rubber on a rubber sheet). Quantum field theory is the quantum mechanics of this infinite system.

The first method used to quantize field theory was the method now called canonical quantization (earlier known as second quantization). This method uses a Hamiltonian formulation of the classical problem. The later technique of Feynman path integrals uses a Lagrangian formulation. Many more methods are now in use; for an overview see the article on quantization.

Canonical quantization for bosons

Suppose we have a system of N bosons which can occupy mutually orthogonal single-particle states and so on. The usual method of writing a multi-particle state is to assign a state to each particle and then impose exchange symmetry. As we have seen, the resulting wavefunction is an unwieldy sum of N! terms. In contrast, in the second quantized approach we will simply list the number of particles in each of the single-particle states, with the understanding that the multi-particle wavefunction is symmetric. To be specific, suppose that N = 3, with one particle in state and two in state. The normal way of writing the wavefunction is

In second quantized form, we write this as

which means "one particle in state 1, two particles in state 2, and zero particles in all the other states."

Though the difference is entirely notational, the latter form makes it easy for us to define creation and annihilation operators, which add and subtract particles from multi-particle states. These creation and annihilation operators are very similar to those defined for the quantum harmonic oscillator, which added and subtracted energy quanta. However, these operators literally create and annihilate particles with a given quantum state. The bosonic annihilation operator a2 and creation operator have the following effects:

We may well ask whether these are operators in the usual quantum mechanical sense, i.e. linear operators acting on an abstract Hilbert space. In fact, the answer is yes: they are operators acting on a kind of expanded Hilbert space, known as a Fock space, composed of the space of a system with no particles (the so-called vacuum state), plus the space of a 1-particle system, plus the space of a 2-particle system, and so forth. Furthermore, the creation and annihilation operators are indeed Hermitian conjugates, which justifies the way we have written them.

The bosonic creation and annihilation operators obey the commutation relation

where δ stands for the Kronecker delta. These are precisely the relations obeyed by the "ladder operators" for an infinite set of independent quantum harmonic oscillators, one for each single-particle state. Adding or removing bosons from each state is therefore analogous to exciting or de-exciting a quantum of energy in a harmonic oscillator.

The final step toward obtaining a quantum field theory is to re-write our original N-particle Hamiltonian in terms of creation and annihilation operators acting on a Fock space. For instance, the Hamiltonian of a field of free (non-interacting) bosons is

where Ek is the energy of the k-th single-particle energy eigenstate. Note that

Second quantization for fermions

It turns out that the creation and annihilation operators for fermions must be defined differently, in order to satisfy the Pauli exclusion principle. For fermions, the occupation numbers Ni can only take on the value 0 or 1, since particles cannot share quantum states. We then define the fermionic annihilation operators c and creation operators by

The fermionic creation and annihilation operators obey an anticommutation relation,

One may notice from this that applying a fermionic creation operator twice gives zero, so it is impossible for the particles to share single-particle states, in accordance with the exclusion principle.

Significance of creation and annihilation operators

When we re-write a Hamiltonian using a Fock space and creation and annihilation operators, as in the previous example, the symbol N, which stands for the total number of particles, drops out. This means that the Hamiltonian is applicable to systems with any number of particles. Of course, in many common situations N is a physically important and perfectly well-defined quantity. For instance, if we are describing a gas of atoms sealed in a box, the number of atoms had better remain a constant at all times. This is certainly true for the above Hamiltonian. Viewing the Hamiltonian as the generator of time evolution, we see that whenever an annihilation operator ak destroys a particle during an infinitesimal time step, the creation operator to the left of it instantly puts it back. Therefore, if we start with a state of N non-interacting particles then we will always have N particles at a later time.

On the other hand, it is often useful to consider quantum states where the particle number is ill-defined, i.e. linear superpositions of vectors from the Fock space that possess different values of N. For instance, it may happen that our bosonic particles can be created or destroyed by interactions with a field of fermions. Denoting the fermionic creation and annihilation operators by and Ck, we could add a "potential energy" term to our Hamiltonian such as:

This describes processes in which a fermion in state k either absorbs or emits a boson, thereby being kicked into a different eigenstate k + q. In fact, this is the expression for the interaction between phonons and conduction electrons in a solid. The interaction between photons and electrons is treated in a similar way; it is a little more complicated, because the role of spin must be taken into account. One thing to notice here is that even if we start out with a fixed number of bosons, we will generally end up with a superposition of states with different numbers of bosons at later times. On the other hand, the number of fermions is conserved in this case.

In condensed matter physics, states with ill-defined particle numbers are also very important for describing the various superfluids. Many of the defining characteristics of a superfluid arise from the notion that its quantum state is a superposition of states with different particle numbers.

Field operators

We can now define field operators that create or destroy a particle at a particular point in space. In particle physics, these are often more convenient to work with than the creation and annihilation operators, because they make it easier to formulate theories that satisfy the demands of relativity.

Single-particle states are usually enumerated in terms of their momenta (as in the particle in a box problem.) We can construct field operators by applying the Fourier transform to the creation and annihilation operators for these states. For example, the bosonic field annihilation operator is

The bosonic field operators obey the commutation relation

where δ(x) stands for the Dirac delta function. As before, the fermionic relations are the same, with the commutators replaced by anticommutators.

It should be emphasized that the field operator is not the same thing as a single-particle wavefunction. The former is an operator acting on the Fock space, and the latter is just a scalar field. However, they are closely related, and are indeed commonly denoted with the same symbol. If we have a Hamiltonian with a space representation, say

where the indices i and j run over all particles, then the field theory Hamiltonian is

This looks remarkably like an expression for the expectation value of the energy, with φ playing the role of the wavefunction. This relationship between the field operators and wavefunctions makes it very easy to formulate field theories starting from space-projected Hamiltonians.

Quantization of classical fields

So far, we have shown how one goes from an ordinary quantum theory to a quantum field theory. There are certain systems for which no ordinary quantum theory exists. These are the "classical" fields, such as the electromagnetic field. There is no such thing as a wavefunction for a single photon in classical electromagnetisim, so a quantum field theory must be formulated right from the start.

The essential difference between an ordinary system of particles and the electromagnetic field is the number of dynamical degrees of freedom. For a system of N particles, there are 3N coordinate variables corresponding to the position of each particle, and 3N conjugate momentum variables. One formulates a classical Hamiltonian using these variables, and obtains a quantum theory by turning the coordinate and position variables into quantum operators, and postulating commutation relations between them such as

For an electromagnetic field, the analogue of the coordinate variables are the values of the electrical potential and the vector potential at every point . This is an uncountable set of variables, because is continuous. This prevents us from postulating the same commutation relation as before. The way out is to replace the Kronecker delta with a Dirac delta function. This ends up giving us a commutation relation exactly like the one for field operators! We therefore end up treating "fields" and "particles" in the same way, using the apparatus of quantum field theory. Only by accident electrons were not regarded as de Broglie waves and photons governed by geometrical optics were not the dominant theory when QFT was developed.

Path integral methods

The axiomatic approach

There have been many attempts to put quantum field theory on a firm mathematical footing by formulating a set of axioms for it. These attempts fall into two broad classes.

The first class of axioms (most notably the Wightman, Osterwalder-Schrader, and Haag-Kastler systems) tried to formalize the physicists' notion of an "operator-valued field" within the context of functional analysis. These axioms enjoyed limited success. It was possible to prove that any QFT satisfying these axioms satisfied certain general theorems, such as the spin-statistics theorem and the PCT theorems. Unfortunately, it proved extraordinarily difficult to show that any realistic field theory (e.g. quantum chromodynamics) satisfied these axioms. Most of the theories which could be treated with these analytic axioms were physically trivial: restricted to low-dimensions and lacking in interesting dynamics.

In the 1980s, a second wave of axioms were proposed. These axioms (associated most closely with Atiyah and Segal, and notably expanded upon by Witten, Borcherds, and Kontsevich) are more geometric in nature, and more closely resemble the path integrals of physics. They have not been exceptionally useful to physicists, as it is still extraordinarily difficult to show that any realistic QFTs satisfy these axioms, but have found many applications in mathematics, particularly in representation theory, algebraic topology, and geometry.

Finding the proper axioms for quantum field theory is still an open and difficult problem in mathematics. In fact, one of the Clay Millennium Prizes offers $1,000,000 to anyone who proves the existence of a mass gap in Yang-Mills theory. It seems likely that we have not yet understood the underlying structures which permit the Feynman path integrals to exist.

Renormalization

Some of the problems and phenomena eventually addressed by renormalization actually appeared earlier in the classical electrodynamics of point particles in the 19th and early 20th century. The basic problem is that the observable properties of an interacting particle cannot be entirely separated from the field that mediates the interaction. The standard classical example is the energy of a charged particle. To cram a finite amount of charge into a single point requires an infinite amount of energy; this manifests itself as the infinite energy of the particle's electric field. The energy density grows to infinity as one gets close to the charge.

A single particle state in quantum field theory incorporates within it multiparticle states. This is most simply demonstrated by examining the evolution of a single particle state in the interaction picture—

Taking the overlap with the initial state, one retains the even powers of HI. These terms are responsible for changing the number of particles during propagation, and are therefore quintessentially a product of quantum field theory. Corrections such as these are incorporated into wave-function renormalization and mass renormalization. Similar corrections to the interaction Hamiltonian, HI, include vertex renormalization, or, in modern language, effective field theory.

History

More details can be found in the article on the history of quantum field theory.

Quantum field theory was created by Dirac when he attempted to quantize the electromagnetic field in the late 1920s. The early development of the field involved Fock, Jordan, Pauli, Heisenberg, Bethe, Tomonaga, Schwinger, Feynman, and Dyson. This phase of development culminated with the construction of the theory of quantum electrodynamics in the 1950s.

Gauge theory was formulated and quantized, leading to the unification of forces embodied in the standard model of particle physics. This effort started in the 1950s with the work of Yang and Mills, was carried on by Martinus Veltman and a host of others during the 1960s and completed during the 1970s by the work of Gerard 't Hooft, Frank Wilczek, David Gross and David Politzer.

Parallel developments in the understanding of phase transitions in condensed matter physics led to the study of the renormalization group. This in turn led to the grand synthesis of theoretical physics which unified theories of particle and condensed matter physics through quantum field theory. This involved the work of Michael Fisher and Leo Kadanoff in the 1970s which led to the seminal reformulation of quantum field theory by Kenneth Wilson.

The study of quantum field theory is alive and flourishing, as are applications of this method to many physical problems. It remains one of the most vital areas of theoretical physics today, providing a common language to many branches of physics.