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What is “Quantum field theory” ?

  By admin | October 18, 2013 - 2:32 pm |
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Quantum field theory

Quantum field theory
Quantum field theory
Quantum field theory
In theoretical physics, quantum field theory is a theoretical framework for constructing quantum mechanical models of subatomic particles in particle physics and quasiparticles in condensed matter physics, by treating a particle as an happy state of an underlying physical field. These happy states are called field quanta. For example, quantum electrodynamics (QED) has one electron field and one photon field, quantum chromodynamics (QCD) has one field for each type of quark, and in condensed matter there is an atomic displacement field that gives rise to phonon particles. Ed Witten describes QFT as “by far” the most difficult theory in modern physics.
In QFT, interactions between particles in quantum mechanics are represented by interaction terms between the underlying fields. QFT interaction terms are similar in spirit to those between electric and magnetic fields in Maxwell’s equations. However unlike the classical fields of Maxwell’s theory, fields in QFT generally exist in quantum superpositions of states and obey the laws of quantum mechanics.
Quantum mechanical systems have a fixed number of particles, with each particle having a small number of degrees of freedom. In contrast, the happy states of a QFT can represent any number of particles. This makes quantum field theories especially useful for describing systems where the particle count/number may change over time.
Because the fields are continuous quantities over space, there exist happy states with arbitrarily large numbers of particles in them, giving QFT systems an effectively infinite number of degrees of freedom. Infinite degrees of freedom can easily lead to divergences of calculated quantities in a straightforward approach to QFT calculations. Techniques such as renormalization of QFT parameters and discretization of spacetime, as in lattice QCD, are used to avoid such infinities and to yield physically meaningful results.
Most theories in modern particle physics are formulated as relativistic quantum field theories, such as QED, QCD, and the Standard Model. In QED the quantum field-theoretic description of the electromagnetic field approximately reproduces Maxwell’s theory of electrodynamics in the low-energy limit, with small non-linear corrections to the Maxwell equations being required due to virtual electron–positron pairs.
In the perturbative approach to quantum field theory, the full field interaction terms are approximated as a perturbative expansion in the number of particles involved. Each term in the expansion can be thought of as forces between particles being mediated by other particles. In QED, the electromagnetic force between two electrons is caused by an exchange of photons. Similarly, intermediate vector bosons mediate the weak force and gluons mediate the strong force in QCD. The notion of a force-mediating particle comes from perturbation theory, and does not make sense in the context of non-perturbative approaches to QFT, such as with bound states.
The gravitational field and the electromagnetic field are the only two fundamental fields in nature that have infinite range and a corresponding classical low-energy limit, which greatly diminishes and hides their “particle-like” excitations. Albert Einstein, in 1905, attributed “particle-like” and discrete exchanges of momenta and energy, characteristic of “field quanta”, to the electromagnetic field. Originally, his principal motivation was to explain the thermodynamics of radiation. Although the photoelectric effect and Compton scattering strongly suggest the existence of the photon, it is now understood that they can be explained without invoking a quantum electromagnetic field; therefore, a more definitive proof of the quantum nature of radiation is now taken up into modern quantum optics as in the antibunching effect. The word “photon” was coined in 1926 by physical chemist Gilbert Newton Lewis.
There is currently no complete quantum theory of the remaining fundamental force, gravity. Many of the proposed theories to describe gravity as a QFT postulate the existence of a graviton particle that mediates the gravitational force. Presumably, the as yet unknown correct quantum field-theoretic treatment of the gravitational field will behave like Einstein’s general theory of relativity in the low-energy limit. Quantum field theory of the fundamental forces itself has been postulated to be the low-energy effective field theory limit of a more fundamental theory such as superstring theory.
The early development of the field involved Dirac, Fock, Pauli, Heisenberg, Bogolyubov. This phase of development culminated with the construction of the theory of quantum electrodynamics in the 1950s.
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 G. Wilson.
A classical field is a function defined over some region of space and time. Two physical phenomena which are described by classical fields are Newtonian gravitation, described by Newtonian gravitational field g, and classical electromagnetism, described by the electric and magnetic fields E and B. Because such fields can in principle take on distinct values at each point in space, they are said to have infinite degrees of freedom.
Classical field theory does not, however, account for the quantum-mechanical aspects of such physical phenomena. For instance, it is known from quantum mechanics that certain aspects of electromagnetism involve discrete particles—photons—rather than continuous fields. The business of quantum field theory is to write down a field that is, like a classical field, a function defined over space and time, but which also accommodates the observations of quantum mechanics. This is a quantum field.
It isn’t immediately clear how to write down such a quantum field, since quantum mechanics has a structure very unlike a field theory. In its most general formulation, quantum mechanics is a theory of abstract operators acting on an abstract state space (Hilbert space), where the observables represent physically observable quantities and the state space represents the possible states of the system under study. For instance, the fundamental observables associated with the motion of a single quantum mechanical particle are the position and momentum operators and. Field theory, in contrast, treats x as a way to index the field rather than as an operator.
There are two common ways of developing a quantum field: the path integral formalism and canonical quantization. The latter of these is pursued in this article.
This equation is Newton’s law of universal gravitation, expressed in differential form in terms of the gravitational potential φ and the mass density ρ. Despite the nomenclature, the “field” under study is the gravitational potential, φ, rather than the gravitational field, g. Similarly, when classical field theory is used to study electromagnetism, the “field” of interest is the electromagnetic four-potential (V/c, A), rather than the electric and magnetic fields E and B.
Quantum field theory uses this same Lagrangian procedure to determine the equations of motion for quantum fields. These equations of motion are then supplemented by commutation relations derived from the canonical quantization procedure described below, thereby incorporating quantum mechanical effects into the behavior of the field.
Here m is the particle’s mass and V(x) is the applied potential. Physical information about the behavior of the particle is extracted from the wavefunction by constructing probability density functions for various quantities; for example, the p.d.f. for the particle’s position is ψ*(x)xψ(x), and the p.d.f. for the particle’s momentum is −iħψ*(x)dψ/dx. This treatment of quantum mechanics, where a particle’s wavefunction evolves against a classical background potential V(x), is sometimes called 1st quantization.
This description of quantum mechanics can be extended to describe the behavior of multiple particles, so long as the number and the type of particles remain fixed. The particles are described by a wavefunction ψ which is governed by an extended version of the Schrxdinger equation.
Often one is interested in the case where N particles are all of the same type. As described in the article on identical particles, this implies that the state of the entire system must be either symmetric (bosons) or antisymmetric (fermions) when the coordinates of its constituent particles are exchanged. This is achieved by using a Slater determinant as the wavefunction of a fermionic system (and a Slater permanent for a bosonic system), which is equivalent to an element of the symmetric or antisymmetric subspace of a tensor product.
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! distinct terms. is a normalizing factor.
There are several shortcomings to the above description of quantum mechanics which are addressed by quantum field theory. First, it is unclear how to extend quantum mechanics to include the effects of special relativity. Attempted replacements for the Schrxdinger equation, such as the Klein–Gordon equation or the Dirac equation, have many unsatisfactory qualities; for instance, they possess energy eigenvalues that extend to –∞, so that there seems to be no easy definition of a ground state. It turns out that such inconsistencies arise from relativistic wavefunctions having a probabilistic interpretation in position space, as probability conservation isn’t a relativistically covariant concept. The 2nd shortcoming, related to the first, is that in quantum mechanics there is no mechanism to describe particle creation and annihilation; this is crucial for describing phenomena such as pair production which result from the conversion between mass and energy according to the relativistic relation E = mc2.
Several other approaches exist, such as the Feynman path integral, which uses a Lagrangian formulation. For an overview of some of these approaches, see the article on quantization.
An N-particle state belongs to a space of states describing systems of N particles. The next step is to combine the individual N-particle state spaces into an extended state space, known as Fock space, which can describe systems of any number of particles. This is composed of the state space of a system with no particles, plus the state space of a 1-particle system, plus the state space of a 2-particle system, and so forth. States describing a definite number of particles are known as Fock states: a general element of Fock space will be a linear combination of Fock states. There is a one-to-one correspondence between the occupation number representation and valid boson states in the Fock space.

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