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 relativistic
quantum 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 particles
and, 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 covariant
concept. 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 relation
allows 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,
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. The way to simplify this problem is to turn it into a quantum field theory.
[edit] Second quantizationMain article: Second quantization
In 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 bosonsFor 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 oscillator
problem, which added and subtracted energy quanta. However, these
operators literally create and annihilate particles of a given quantum
state. The bosonic annihilation operator
a2 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 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 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
Ek is the energy of the
k-th single-particle energy eigenstate. Note that
[edit] Second quantization of fermionsIt 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
Ni 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.
- Feynman, R.P. (2006) [1985]. QED: The Strange Theory of Light and Matter. Princeton University Press. ISBN 0691125759.
- Gribbin, J. (1998). Q is for Quantum: Particle Physics from A to Z. Weidenfeld & Nicolson. ISBN 0297817523.
- Schumm, Bruce A. (2004) Deep Down Things. Johns Hopkins Univ. Press. Chpt. 4.
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 φ4-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.
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