**Quantum field theory** (

**QFT**)

^{[1]} provides a theoretical framework for constructing

quantum mechanical models of systems classically described by

fields or (especially in a condensed matter context)

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

relativisticquantum field theories. Quantum field theories are used in many

circumstances, especially those where the number of particles

fluctuates—for example, in the

BCS theory of

superconductivity.

In

perturbative quantum field theory, the forces between particles are mediated by other particles. The

electromagnetic force between two

electrons is caused by an exchange of

photons.

Intermediate vector bosons mediate the

weak force and

gluons mediate the

strong force. There is currently no complete quantum theory of the remaining fundamental force,

gravity, but many of the

proposed theories postulate the existence of a

graviton particle which mediates it. These force-carrying particles are

virtual particlesand, by definition, cannot be detected while carrying the force,

because such detection will imply that the force is not being carried.

In addition, the notion of "force mediating particle" comes from

perturbation theory, and thus does not make sense in a context of bound

states.

In QFT photons are not thought of as 'little billiard balls', they are considered to be

field quanta - necessarily chunked ripples in a field that 'look like' particles.

Fermions,

like the electron, can also be described as ripples in a field, where

each kind of fermion has its own field. In summary, the classical

visualisation of "everything is particles and fields", in quantum field

theory, resolves into "everything is particles", which then resolves

into "everything is fields". In the end, particles are regarded as

excited states of a field (field quanta).

In ordinary quantum mechanics, the time-dependent one-dimensional

Schrödinger equation describing the time evolution of the quantum state of a single non-relativistic particle is

where

*m* is the particle's

mass,

*V* is the applied

potential, and

denotes the

quantum state (we are using

bra-ket notation).

We wish to consider how this problem generalizes to

*N* particles. There are two motivations for studying the many-particle problem. The first is a straightforward need in

condensed matter physics, where typically the number of particles is on the order of

Avogadro's number (6.0221415 x 10

^{23}). The second motivation for the many-particle problem arises from

particle physics and the desire to incorporate the effects of

special relativity. If one attempts to include the relativistic

rest energy into the above equation (in quantum mechanics where position is an observable), the result is either 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.

It turns out that such inconsistencies arise from relativistic

wavefunctions having a probabilistic interpretation in position space,

as probability conservation is not a relativistically

covariantconcept. In quantum field theory, unlike in quantum mechanics, position

is not an observable, and thus, one does not need the concept of a

position-space probability density. For quantum fields whose

interaction can be treated perturbatively, this is equivalent to

neglecting the possibility of dynamically creating or destroying

particles, which is a crucial aspect of relativistic quantum theory.

Einstein's famous

mass-energy relationallows for the possibility 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. This suggests that a consistent relativistic quantum theory should be able to describe many-particle dynamics.

Furthermore, we will assume that the

*N* particles are

indistinguishable. 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. These

multi-particle states are rather complicated to write. For example, the

general quantum state of a system of

*N* bosons is written as

where

are the single-particle states,

*N*_{j} 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. The way to simplify this problem is to turn it into a quantum field theory.

**[edit] Second quantization**Main article:

Second quantizationIn this section, we will describe a method for constructing a quantum field theory called

**second quantization**.

This basically involves choosing a way to index the quantum mechanical

degrees of freedom in the space of multiple identical-particle states.

It is based on the

Hamiltonian formulation of quantum mechanics; several other approaches exist, such as the

Feynman path integral^{[3]}, which uses a

Lagrangian formulation. For an overview, see the article on

quantization.

**[edit] Second quantization of bosons**For simplicity, we will first discuss second quantization for

bosons, which form perfectly symmetric quantum states. Let us denote the mutually orthogonal single-particle states by

and so on. For example, the 3-particle state with one particle in state

and two in state

is

The first step in second quantization is to express such quantum states in terms of

**occupation numbers**, by listing the number of particles occupying each of the single-particle states

etc. This is simply another way of labelling the states. For instance, the above 3-particle state is denoted as

The next step is to expand the

*N*-particle state space to include the state spaces for all possible values of

*N*. This extended state space, known as a

Fock space, is composed of the state space of a system with no particles (the so-called

vacuum state),

plus the state space of a 1-particle system, plus the state space of a

2-particle system, and so forth. It is easy to see that there is a

one-to-one correspondence between the occupation number representation

and valid boson states in the Fock space.

At this point, the quantum mechanical system has become a quantum

field in the sense we described above. The field's elementary degrees

of freedom are the occupation numbers, and each occupation number is

indexed by a number

, indicating which of the single-particle states

it refers to.

The properties of this quantum field can be explored by defining

creation and annihilation operators, which add and subtract particles. They are analogous to "ladder operators" in the

quantum harmonic oscillatorproblem, which added and subtracted energy quanta. However, these

operators literally create and annihilate particles of a given quantum

state. The bosonic annihilation operator

*a*_{2} and creation operator

have the following effects:

It can be shown that these are operators in the usual quantum mechanical sense, i.e.

linear operators acting on the Fock space. Furthermore, they are indeed

Hermitian conjugates, which justifies the way we have written them. They can be shown to obey the

commutation relationwhere 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

Hamiltonian of the quantum field (which, through the

Schrödinger equation,

determines its dynamics) can be written in terms of creation and

annihilation operators. For instance, the Hamiltonian of a field of

free (non-interacting) bosons is

where

*E*_{k} is the energy of the

*k*-th single-particle energy eigenstate. Note that

**[edit] Second quantization of fermions**It turns out that a different definition of creation and annihilation must be used for describing

fermions. According to the

Pauli exclusion principle, fermions cannot share quantum states, so their occupation numbers

*N*_{i} can only take on the value 0 or 1. The fermionic annihilation operators

*c* and creation operators

are defined by their actions on a Fock state thus

These 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.

Feynman, R.P. (2001) [1964].

*The Character of Physical Law*.

MIT Press.

ISBN 0262560038.

**Introductory texts**:

- Bogoliubov, N.; Shirkov, D. (1982).
*Quantum Fields*. Benjamin-Cummings. ISBN 0805309837.

- Frampton, P.H. (2000).
*Gauge Field Theories**. Frontiers in Physics (2nd ed.). Wiley.*

- Greiner, W; Müller, B. (2000).
*Gauge Theory of Weak Interactions*. Springer. ISBN 3-540-67672-4.

- Itzykson, C.; Zuber, J.-B. (1980).
*Quantum Field Theory*. McGraw-Hill. ISBN 0-07-032071-3.

- Kane, G.L. (1987).
*Modern Elementary Particle Physics*. Perseus Books. ISBN 0-201-11749-5.

- Kleinert, H.; Schulte-Frohlinde, Verena (2001).
*Critical Properties of φ*^{4}-Theories. World Scientific. ISBN 981-02-4658-7. http://users.physik.fu-berlin.de/~kleinert/re.html#B6.

- Kleinert, H. (2008).
*Multivalued Fields in Condensed Matter, Electrodynamics, and Gravitation*. World Scientific. ISBN 978-981-279-170-2. http://users.physik.fu-berlin.de/~kleinert/public_html/kleiner_reb11/psfiles/mvf.pdf.

- Loudon, R (1983).
*The Quantum Theory of Light*. Oxford University Press. ISBN 0-19-851155-8.

- Mandl, F.; Shaw, G. (1993).
*Quantum Field Theory*. John Wiley & Sons. ISBN 0-0471-94186-7.

- Peskin, M.; Schroeder, D. (1995).
*An Introduction to Quantum Field Theory*. Westview Press. ISBN 0-201-50397-2.

- Ryder, L.H. (1985).
*Quantum Field Theory*. Cambridge University Press. ISBN 0-521-33859-X.

- Srednicki, Mark (2007)
*Quantum Field Theory.* Cambridge Univ. Press.

- Yndurain, F.J. (1996).
*Relativistic Quantum Mechanics and Introduction to Field Theory* (1st ed.). Springer. ISBN 978-3540604532.

- Zee, A. (2003).
*Quantum Field Theory in a Nutshell*. Princeton University Press. ISBN ISBN 0-691-01019-6.