Everything about Relativistic Quantum Field Theory totally explained
Quantum field theory (QFT) is a theoretical framework for constructing
quantum mechanical models of
field-like systems or, equivalently, of
many-body systems. It is widely used in
particle physics and
condensed matter physics. Most theories in modern particle physics, including the
Standard Model of elementary particles and their interactions, are formulated as
relativistic quantum field theories. In condensed matter physics, quantum field theories are used in many circumstances, especially those where the number of particles is allowed to fluctuate—for example, in the
BCS theory of
superconductivity.
History
1920s from the problem of creating a
quantum mechanical theory of the
electromagnetic field. In
1926,
Max Born,
Pascual Jordan, and
Werner Heisenberg constructed such a theory by expressing the field's internal
degrees of freedom as an infinite set of
harmonic oscillators and by employing the usual procedure for quantizing those oscillators (
canonical quantization). This theory assumed that no electric charges or currents were present and today would be called a
free field theory. The first reasonably complete theory of
quantum electrodynamics, which included both the electromagnetic field and electrically charged matter (specifically,
electrons) as quantum mechanical objects, was created by
Paul Dirac in
1927. This quantum field theory could be used to model important processes such as the emission of a
photon by an electron dropping into a
quantum state of lower energy, a process in which the
number of particles changes — one atom in the initial state becomes an atom plus a
photon in the final state. It is now understood that the ability to describe such processes is one of the most important features of quantum field theory.
It was evident from the beginning that a proper quantum treatment of the electromagnetic field had to somehow incorporate
Einstein's relativity theory, which had after all grown out of the study of
classical electromagnetism. This need to
put together relativity and quantum mechanics was the second major motivation in the development of quantum field theory.
Pascual Jordan and
Wolfgang Pauli showed in
1928 that quantum fields could be made to behave in the way predicted by
special relativity during
coordinate transformations (specifically, they showed that the field
commutators were
Lorentz invariant), and in
1933 Niels Bohr and
Leon Rosenfeld showed that this result could be interpreted as a limitation on the ability to measure fields at
space-like separations, exactly as required by relativity. A further boost for quantum field theory came with the discovery of the
Dirac equation, a single-particle equation obeying both relativity and quantum mechanics, when it was shown that several of its undesirable properties (such as negative-energy states) could be eliminated by reformulating the Dirac equation as a quantum field theory. This work was performed by
Wendell Furry,
Robert Oppenheimer,
Vladimir Fock, and others.
The third thread in the development of quantum field theory was the need to
handle the statistics of many-particle systems consistently and with ease. In
1927, Jordan tried to extend the canonical quantization of fields to the many-body wavefunctions of
identical particles, a procedure that's sometimes called
second quantization. In
1928, Jordan and
Eugene Wigner found that the quantum field describing electrons, or other
fermions, had to be expanded using anti-commuting creation and annihilation operators due to the
Pauli exclusion principle. This thread of development was incorporated into
many-body theory, and strongly influenced
condensed matter physics and
nuclear physics.
Despite its early successes, quantum field theory was plagued by several serious theoretical difficulties. Many seemingly-innocuous physical quantities, such as the energy shift of electron states due to the presence of the electromagnetic field, gave infinity — a nonsensical result — when computed using quantum field theory. This "divergence problem" was solved during the
1940s by
Bethe,
Tomonaga,
Schwinger,
Feynman, and
Dyson, through the procedure known as
renormalization. This phase of development culminated with the construction of the modern theory of
quantum electrodynamics (QED). Beginning in the
1950s with the work of
Yang and
Mills, QED was generalized to a class of quantum field theories known as
gauge theories. The
1960s and
1970s saw the formulation of a gauge theory now known as the
Standard Model of
particle physics, which describes all known elementary particles and the interactions between them. The weak interaction part of the standard model was formulated by
Sheldon Glashow, with the
Higgs mechanism added by
Steven Weinberg and
Abdus Salam. The theory was shown to be consistent by
Gerardus 't Hooft and
Martinus Veltman.
Also during the
1970s, parallel developments in the study of
phase transitions in
condensed matter physics led
Leo Kadanoff,
Michael Fisher and
Kenneth Wilson (extending work of
Ernst Stueckelberg,
Andre Peterman,
Murray Gell-Mann and
Francis Low) to a set of ideas and methods known as the
renormalization group. By providing a better physical understanding of the renormalization procedure invented in the
1940s, the renormalization group sparked what has been called the "grand synthesis" of theoretical physics, uniting the quantum field theoretical techniques used in particle physics and condensed matter physics into a single theoretical framework.
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.
Principles of quantum field theory
Classical fields and quantum fields
Quantum mechanics, in its most general formulation, is a theory of abstract
operators (observables) 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. Furthermore, each observable
corresponds, in a technical sense, to the classical idea of a
degree of freedom. For instance, the fundamental observables associated with the motion of a single quantum mechanical particle are the position and momentum operators
,
where
and
denotes the bosonic creation and annihilation operators,
and
denotes the fermionic creation and annihilation operators, and
is a parameter that describes the strength of the interaction. This "interaction term" describes processes in which a fermion in state
either absorbs or emits a boson, thereby being kicked into a different eigenstate
. (In fact, this type of Hamiltonian is used to describe interaction between
conduction electrons and
phonons in
metals. The interaction between electrons and
photons is treated in a similar way, but is a little more complicated because the role of
spin must be taken into account.) One thing to notice here's that even if we start out with a fixed number of bosons, we'll typically end up with a superposition of states with different numbers of bosons at later times. The number of fermions, however, is conserved in this case.
In
condensed matter physics, states with ill-defined particle numbers are particularly 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.
Axiomatic approaches
The preceding description of quantum field theory follows the spirit in which most
physicists approach the subject. However, it isn't
mathematically rigorous. Over the past several decades, 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, first proposed during the
1950s, include the
Wightman,
Osterwalder-Schrader, and
Haag-Kastler systems. They attempted to formalize the physicists' notion of an "operator-valued field" within the context of
functional analysis, and enjoyed limited success. It was possible to prove that any quantum field theory satisfying these axioms satisfied certain general theorems, such as the
spin-statistics theorem and the
CPT theorem. Unfortunately, it proved extraordinarily difficult to show that any realistic field theory, including the
Standard Model, satisfied these axioms. Most of the theories that could be treated with these analytic axioms were physically trivial, being restricted to low-dimensions and lacking interesting dynamics. The construction of theories satisfying one of these sets of axioms falls in the field of
constructive quantum field theory. Important work was done in this area in the
1970s by Segal, Glimm, Jaffe and others.
During the
1980s, a second set of axioms based on
geometric ideas was proposed. This line of investigation, which restricts its attention to a particular class of quantum field theories known as
topological quantum field theories, is associated most closely with
Michael Atiyah and
Graeme Segal, and was notably expanded upon by
Edward Witten,
Richard Borcherds, and
Maxim Kontsevich. However, most physically-relevant quantum field theories, such as the
Standard Model, are not topological quantum field theories; the quantum field theory of the
fractional quantum Hall effect is a notable exception. The main impact of axiomatic topological quantum field theory has been on mathematics, with important applications in
representation theory,
algebraic topology, and
differential geometry.
Finding the proper axioms for quantum field theory is still an open and difficult problem in mathematics. One of the
Millennium Prize Problems—proving the existence of a
mass gap in Yang-Mills theory—is linked to this issue.
Phenomena associated with quantum field theory
In the previous part of the article, we described the most general properties of quantum field theories. Some of the quantum field theories studied in various fields of theoretical physics possess additional special properties, such as renormalizability, gauge symmetry, and supersymmetry. These are described in the following sections.
Renormalization
perturbative shift in the energy of an electron due to the presence of the electromagnetic field, give infinite results. Many of these problems are related to failures in
classical electrodynamics that were identified (but unsolved) as far back as the
19th century, and they basically stem from the fact that many of the supposedly "intrinsic" properties of an electron are tied to the electromagnetic field with which it interacts. To illustrate this, recall from the preceding discussion that the interaction Hamiltonian between two quantum fields, such as the electron field and the electromagnetic field, need not conserve particle number. Thus, even if we start out with a single electron and no photons, the quantum state will rapidly evolve into a superposition of states that can include one or more photons. Therefore, the energy carried by that "single" electron—its
self energy—is not simply the "bare" value, but also includes the energy contained in an attendant cloud of photons. When this self energy is computed, one finds that the contribution of photons possessing arbitrarily high energies (or, equivalently, arbitrarily short
wavelengths) leads to a formally infinite value.
The solution to this problem, first given by
Julian Schwinger, is called
renormalization. The idea is to impose a "
cutoff" for the photonic contribution, for example by postulating that photons can't possess energies above some extremely high value. Any quantity we wish to compute, such as the rest energy, is now finite but dependent on the cutoff. We then recast the result in terms of physically-observable quantities such as the observed electron mass, instead of unobservable quantities such as the cutoff energy and the bare electron mass. The final result is independent of all details of the cutoff procedure, including the value of the cutoff energy, provided the relevant processes occur at energies far below the cutoff.
The renormalization procedure only works for a certain class of quantum field theories, called
renormalizable quantum field theories. The
Standard Model of
particle physics is renormalizable, and so are its component theories (
quantum electrodynamics/
electroweak theory and
quantum chromodynamics). According to the theory of the
renormalization group, each renormalizable theory is a
unique low-energy limit (for example a so-called "
effective field theory") for a broad range of high-energy theories. Renormalizable theories are therefore independent of the precise nature of the underlying high-energy phenomena.
Gauge freedom
A
gauge theory is a theory that admits a
symmetry with a local parameter. For example, in every
quantum theory the global
phase of the
wave function is arbitrary and doesn't represent something physical. Consequently, the theory is invariant under a global change of phases (adding a constant to the phase of all wave functions, everywhere); this is a
global symmetry. In
quantum electrodynamics, the theory is also invariant under a
local change of phase, that's - one may shift the phase of all
wave functions so that the shift may be different at every point in
space-time. This is a
local symmetry. However, in order for a well-defined
derivative operator to exist, one must introduce a new
field, the
gauge field, which also transforms in order for the local change of variables (the phase in our example) not to affect the derivative. In
quantum electrodynamics this
gauge field is the
electromagnetic field. The change of local gauge of variables is termed
gauge transformation.
In quantum field theory the excitations of fields represent
particles. The particle associated with excitations of the
gauge field is the
gauge boson, which is the
photon in the case of
quantum electrodynamics.
The
degrees of freedom in quantum field theory are local fluctuations of the fields. The existence of a
gauge symmetry reduces the number of degrees of freedom, simply because some fluctuations of the fields can be transformed to zero by
gauge transformations, so they're equivalent to having no fluctuations at all, and they therefore have no physical meaning. Such fluctuations are usually called "non-physical degrees of freedom" or
gauge artifacts; usually some of them have a negative
norm, making them inadequate for a consistent theory. Therefore, if a classical field theory has a gauge symmetry, then its quantized version (for example the corresponding quantum field theory) will have this symmetry as well. In other words, a gauge symmetry can't have a quantum
anomaly. If a gauge symmetry is
anomalous (for example not kept in the quantum theory) then the theory is non-consistent: for example, in
quantum electrodynamics, had there been a
gauge anomaly, this would require the appearance of
photons with
longitudinal polarization and
polarization in the time direction, the latter having a negative
norm, rendering the theory inconsistent; another possibility would be for these photons to appear only in intermediate processes but not in the final products of any interaction, making the theory non
unitary and again inconsistent (see
optical theorem).
In general, the
gauge transformations of a theory consist several different transformations, which may not be
commutative. These transformations are together described by a mathematical object known as a
gauge group.
Infinitesimal gauge transformations are the
gauge group generators. Therefore the number of
gauge bosons is the group
dimension (for example number of generators forming a
basis).
All the
fundamental interactions in nature are described by
gauge theories. These are:
Supersymmetry
Supersymmetry assumes that every fundamental
fermion has a superpartner that's a
boson and vice versa. It was introduced in order to solve the so-called
Hierarchy Problem, that is, to explain why particles not protected by any symmetry (like the
Higgs boson) don't receive radiative corrections to its mass driving it to the larger scales (GUT, Planck...). It was soon realized that supersymmetry has other interesting properties: its gauged version is an extension of general relativity (
Supergravity), and it's a key ingredient for the consistency of
string theory.
The way supersymmetry protects the hierarchies is the following: since for every particle there's a superpartner with the same mass, any loop in a radiative correction is cancelled by the loop corresponding to its superpartner, rendering the theory UV finite.
Since no superpartners have yet been observed, if supersymmetry exists it must be broken (through a so-called soft term, which breaks supersymmetry without ruining its helpful features). The simplest models of this breaking require that the energy of the superpartners not be too high; in these cases, supersymmetry is expected to be observed by experiments at the
Large Hadron Collider.
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