The most sophisticated form of quantum theory, in which all matter and force particles are expressed as sums over simple waves. It is essential for understanding the processes in which particles are created or destroyed, as when electrons and positrons annihilate at the same time, and is applied to solid state and particle physics. Quantum electrodynamics, quantum chromodynamics, and the GlashowWeinbergSalam theory are all quantum field theories.
Quantum field theory (QFT) is the quantum theory of 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 energy radiated by an atom when it dropped from one quantum state to another of lower energy. 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. Quantum field theory followed unavoidably from a quantum treatment of the only known classical field, viz.
It was obvious from the beginning that the quantum treatment of the electromagnetic field required a proper treatment of relativity. 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.
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. 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).
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...).
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. 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. 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. 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. 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. 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. Each normal mode oscillation of the field is interpreted as a particle with frequency f. 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. 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). 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. 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. 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.
The bosonic creation and annihilation operators obey the commutation relation
where δ stands for the Kronecker delta.
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. 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. 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. For instance, it may happen that our bosonic particles can be created or destroyed by interactions with a field of fermions.
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.
It should be emphasized that the field operator is not the same thing as a single-particle wavefunction. 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 . We therefore end up treating "fields" and "particles" in the same way, using the apparatus of quantum field theory.
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.
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. Unfortunately, it proved extraordinarily difficult to show that any realistic field theory (e.g. Constructive quantum field theory is the construction of theories which satisfy one of these sets of axioms.
Finding the proper axioms for quantum field theory is still an open and difficult problem in mathematics.
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. this manifests itself as the infinite energy of the particle's electric field.
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. Similar corrections to the interaction Hamiltonian, HI, include vertex renormalization, or, in modern language, effective field theory.
Gauge theories
A gauge theory is a theory which admits a symmetry with a local parameter. For example, in every quantum theory the global phase of the wave function is arbitrary and does not represent something physical, so the theory is invariant under a global change of phases (adding a constant to the phase of all wave functions, everywhere); In quantum electrodynamics, the theory is also invariant under a local change of phase, that is - one may shift the phase of all wave functions so that in every point in space-time the shift is different. 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.
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 are equivalent to having no fluctuations at all, and they therefore have no physical meaning. Therefore, if a classical field theory has a gauge symmetry, then its quantized version (i.e. the corresponding quantum field theory) will have this symmetry as well. 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;
In general, the gauge transformations of a theory consist several different transformations, which may not be commutative.
The way supersymmetry protects the hierarchies is the following: since for every particle there is 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.
Beyond local 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. 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 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. Quantum Field Theory in a Nutshell, Princeton University Press (2003) [ISBN 0-691-01019-6]. Peskin, M and Schroeder, D. ;An Introduction to Quantum Field Theory (Westview Press, 1995) [ISBN 0-201-50397-2] Weinberg, Steven ;
User Comments Add a comment…