Quantum field theory

Quantum field theory (QFT) is the application of quantum mechanics to 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.

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. Here is how it is:

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1. Each normal mode oscillation of the field is interpreted as a particle with frequency f.

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2. The quantum number n of each normal mode (which can be thought of as a harmonic oscillator) is interpreted as the number of particles.

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The energy associated with the particle is therefore = (n+1/2)hf.

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With some thought, one may similarly associate momenta and position of particles with observables of the field.

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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.

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Two caveats should be made before proceeding further:

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• Each of these "particles" obeys the usual uncertainty principle of quantum mechanics. The "field" is an operator defined at each point of spacetime.
• 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). This method uses a Hamiltonian formulation of the classical problem. The later technique of Feynman path integrals uses a Lagrangian formulation. Many more methods are now in use; for an overview see the article on quantization.

Related Topics:
Canonical quantization - Hamiltonian - Feynman path integral - Lagrangian - Quantization

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Canonical quantization for bosons

Suppose we have a system of N bosons which can occupy mutually orthogonal single-particle states |phi_1 ang, |phi_2 ang, |phi_3 ang, and so on. The usual method of writing a multi-particle state is to assign a state to each particle and then impose exchange symmetry. As we have seen, the resulting wavefunction is an unwieldy sum of N! terms. In contrast, in the second quantized approach we will simply list the number of particles in each of the single-particle states, with the understanding that the multi-particle wavefunction is symmetric. To be specific, suppose that N=3, with one particle in state |phi_1 ang and two in state|phi_2 ang. The normal way of writing the wavefunction is

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: rac{1}{sqrt{3}} left[ |phi_1 ang |phi_2 ang

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Second quantization for fermions

It turns out that the creation and annihilation operators for fermions must be defined differently, in order to satisfy the Pauli exclusion principle. For fermions, the occupation numbers N_i 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 c^dagger by

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: c_j | N_1, N_2, cdots, N_j = 0, cdots angle = 0

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: c_j | N_1, N_2, cdots, N_j = 1, cdots angle = (-1)^{(N_1 + cdots + N_{j-1})} | N_1, N_2, cdots, N_j = 0, cdots angle

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: c_j^dagger | N_1, N_2, cdots, N_j = 0, cdots angle = (-1)^{(N_1 + cdots + N_{j-1})} | N_1, N_2, cdots, N_j = 1, cdots angle

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: c_j^dagger | N_1, N_2, cdots, N_j = 1, cdots angle = 0

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The fermionic creation and annihilation operators obey an anticommutation relation,

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:

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left{c_i , c_j ight} = 0 quad,quad

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left{c_i^dagger , c_j^dagger ight} = 0 quad,quad

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left{c_i , c_j^dagger ight} = delta_{ij}

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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.

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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. This means that the Hamiltonian is applicable to systems with any number of particles. Of course, in many common situations N is a physically important and perfectly well-defined quantity. For instance, if we are describing a gas of atoms sealed in a box, the number of atoms had better remain a constant at all times. This is certainly true for the above Hamiltonian. Viewing the Hamiltonian as the generator of time evolution, we see that whenever an annihilation operator a_k destroys a particle during an infinitesimal time step, the creation operator a_k^dagger to the left of it instantly puts it back. Therefore, if we start with a state of N non-interacting particles then we will always have N particles at a later time.

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On the other hand, it is often useful to consider quantum states where the particle number is ill-defined, i.e. linear superpositions of vectors from the Fock space that possess different values of N. For instance, it may happen that our bosonic particles can be created or destroyed by interactions with a field of fermions. Denoting the fermionic creation and annihilation operators by c_k^dagger and c_k, we could add a "potential energy" term to our Hamiltonian such as:

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:V = sum_{k,q} V_q (a_q + a_{-q}^dagger) c_{k+q}^dagger c_k

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This describes processes in which a fermion in state k either absorbs or emits a boson, thereby being kicked into a different eigenstate k+q. In fact, this is the expression for the interaction between phonons and conduction electrons in a solid. The interaction between photons and electrons is treated in a similar way; it is a little more complicated, because the role of spin must be taken into account. One thing to notice here is that even if we start out with a fixed number of bosons, we will generally end up with a superposition of states with different numbers of bosons at later times. On the other hand, the number of fermions is conserved in this case.

Related Topics:
Phonon - Conduction electron - Solid - Spin

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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.

Related Topics:
Condensed matter physics - Superfluid

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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.

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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 phi(mathbf{r}) is

Related Topics:
Momenta - Particle in a box - Fourier transform

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:phi(mathbf{r}) equiv sum_{i} e^{imathbf{k}_icdot mathbf{r}} a_{i}

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The bosonic field operators obey the commutation relation

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:

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left = 0 quad,quad

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left = 0 quad,quad

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left = delta^3(mathbf{r} - mathbf{r'})

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where delta(x) stands for the Dirac delta function. As before, the fermionic relations are the same, with the commutators replaced by anticommutators.

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It should be emphasized that the field operator is not the same thing as a single-particle wavefunction. The former is an operator acting on the Fock space, and the latter is just a scalar field. However, they are closely related, and are indeed commonly denoted with the same symbol. If we have a Hamiltonian with a space representation, say

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:H = - rac{hbar^2}{2m} sum_i abla_i^2 + sum_{i < j} U(|mathbf{r}_i - mathbf{r}_j|)

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where the indices i and j run over all particles, then the field theory Hamiltonian is

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:H = - rac{hbar^2}{2m} int d^3!r ; phi(mathbf{r})^dagger abla^2 phi(mathbf{r}) + int!d^3!r int!d^3!r' ; phi(mathbf{r})^dagger phi(mathbf{r}')^dagger U(|mathbf{r} - mathbf{r}'|) phi(mathbf{r'}) phi(mathbf{r})

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This looks remarkably like an expression for the expectation value of the energy, with phi 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.

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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.

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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

Related Topics:
Classical Hamiltonian - Commutation relations

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:left = delta_{ij}

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For an electromagnetic field, the analogue of the coordinate variables are the values of the electrical potential phi(mathbf{x}) and the vector potential mathbf{A}(mathbf{x}) at every point mathbf{x}. This is an uncountable set of variables, because mathbf{x} is continuous. This prevents us from postulating the same commutation relation as before. The way out is to replace the Kronecker delta with a Dirac delta function. This ends up giving us a commutation relation exactly like the one for field operators! We therefore end up treating "fields" and "particles" in the same way, using the apparatus of quantum field theory.

Related Topics:
Electrical potential - Vector potential - Uncountable set

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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 most prominent of these are the Wightman axioms and the Haag-Kastler axioms.

Related Topics:
Axiom - Wightman axioms - Haag-Kastler axioms

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The classic results gained from the axiomatic approach are the PCT Theorem (stating that the combination of parity, time and charge inversion is an unbroken symmetry) and the spin-statistics theorem (stating that particles of integer valued spin follow the Bose-Einstein statistics and particles of half-integer spin follow the Fermi statistics).

Related Topics:
PCT Theorem - Spin-statistics theorem - Bose-Einstein statistics - Fermi statistics

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