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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 particular, the quantum theory of the electromagnetic field, known as quantum electrodynamics, is one of the most well-tested and successful theories in physics. The fundamentals of quantum field theory were developed between the late 1920s and the 1950s, notably by Dirac, Fock, Pauli, Tomonaga, Schwinger, Feynman, and Dyson.

Contents

1 Shortcomings of ordinary quantum mechanics

2 Quantum fields

2.1 Second quantization
2.2 Significance of creation and annihilation operators
2.3 Field operators
2.4 Quantization of classical fields

3 Axiomatic Approach

4 See also

5 Suggested reading

Shortcomings of ordinary quantum mechanics

Quantum field theory corrects several deficiencies of ordinary quantum mechanics, which we will briefly discuss. The Schrödinger equation, in its most commonly encountered form, is

$\left[ \frac{|\mathbf{p}|^2}{2m} + V(\mathbf{r}) \right] |\psi(t)\rang = i \hbar \frac{\partial}{\partial t} |\psi(t)\rang$

where |ψ> denotes the quantum state of a particle with mass m, acted on by a potential energy V.

There are two problems with this equation. Firstly, it is not relativistic, reducing to classical mechanics rather than relativistic mechanics in the correspondence limit. To see this, we note that the first term on the left is only the classical kinetic energy p²/2m, with the rest energy mc² omitted. It is possible to modify the Schrödinger equation to include the rest energy, resulting in the Klein-Gordon equation or the Dirac equation. However, these equations have many unsatisfactory qualities; for instance, they possess energy spectra which extend to -∞, so that there is no ground state. Such inconsistencies occur because these equations neglect the possibility of dynamically creating or destroying particles, which is a crucial aspect of relativity. 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. For example, an electron and a positron can annihilate each other to create photons. Such processes must be accounted for in a truly relativistic quantum theory.

The second problem occurs when we seek to extend the equation to large numbers of particles. 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. These multi-particle states are extremely complicated to write. For example, the general quantum state of a system of N bosons is written as

$|\phi_1 \cdots \phi_N \rang = \sqrt{\frac{\prod_j N_j}{N!}} \sum_{p} |\phi_{p(1)}\rang \cdots |\phi_{p(N)} \rang$

where |φ_i> 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.

Quantum fields

Second quantization

Both of the above problems are resolved by moving our attention from a fixed set of particles to a quantum field. The procedure by which quantum fields are constructed from individual particles was introduced by Dirac, and is (for historical reasons) known as second quantization.

We should mention two possible points of confusion. Firstly, the aforementioned "field" and "particle" descriptions do not refer to wave-particle duality. By "particle", we refer to entities which possess both wave and point-particle properties in the usual quantum mechanical sense; for example, these "particles" are generally not located at a fixed point, but have a certain probability of being found at each position in space. What we refer to as a "field" is an entity existing at every point in space, which regulates the creation and annihilation of the particles. Secondly, quantum field theory is essentially quantum mechanics, and not a replacement for quantum mechanics. Like any quantum system, a quantum field possesses a Hamiltonian H (albeit one that is more complicated than typical single-particle Hamiltonians), and obeys the usual Schrödinger equation

$H \left| \psi (t) \right\rangle = i \hbar {\partial\over\partial t} \left| \psi (t) \right\rangle$

(Quantum field theories are often formulated in terms of a Lagrangian, for reasons of convenience. However, the Lagrangian and Hamiltonian formulations are believed to be equivalent.)

In second quantization, we make use of particle indistinguishability by specifying multi-particle wavefunctions in terms of single-particle occupation numbers. For example, suppose we have a system of N bosons which can occupy mutually orthogonal single-particle states |φ₁>, |φ₂>, |φ₃>, 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 |φ₁> and two in state |φ₂>. The normal way of writing the wavefunction is

$\frac{1}{\sqrt{3}} \left[ |\phi_1\rang |\phi_2\rang |\phi_2\rang + |\phi_2\rang |\phi_1\rang |\phi_2\rang + |\phi_2\rang |\phi_2\rang |\phi_1\rang \right]$

In second quantized form, we write this as

$|1, 2, 0, 0, 0, \cdots \rangle$

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. These creation and annihilation operators are very similar to those defined for the quantum harmonic oscillator, which added and subtracted energy quanta. However, these operators literally create and annihilate particles with a given quantum state. For example, the annihilation operator a₂ and the creation operator a₂^† have the following effects:

$a_2 \| N_1, N_2, N_3, \cdots \rangle \equiv$	$\sqrt{N_2}$	$\|N_1, (N_2 - 1), N_3, \cdots \rangle$
$a_2^\dagger \| N_1, N_2, N_3, \cdots \rangle \equiv$	$\sqrt{N_2 + 1}$	$\| N_1, (N_2 + 1), N_3, \cdots \rangle$

We may well ask whether these are operators in the usual quantum mechanical sense, i.e. linear operators acting on an abstract Hilbert space. 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. Furthermore, the creation and annihilation operators are indeed Hermitian conjugates, which justifies the way we have written them.

The creation and annihilation operators obey the commutation relation

$\left[a_i , a_j \right] = 0 \quad,\quad \left[a_i^\dagger , a_j^\dagger \right] = 0 \quad,\quad \left[a_i , a_j^\dagger \right] = \delta_{ij}$

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 creation and annihilation operators for fermions obey an anticommutation relation,

$\left\{c_i , c_j \right\} = 0 \quad,\quad \left\{c_i^\dagger , c_j^\dagger \right\} = 0 \quad,\quad \left\{c_i , c_j^\dagger \right\} = \delta_{ij}$

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 Pauli exclusion principle.

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

$H = \sum_k E_k \, a^\dagger_k \,a_k$

where E_k is the energy of the k-th single-particle energy eigenstate.

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^† 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.

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^† and c_k, we could add a "potential energy" term to our Hamiltonian such as:

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

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.

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 fact 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 φ(r) is

$\phi(\mathbf{r}) \equiv \sum_{i} e^{i\mathbf{k}_i\cdot \mathbf{r}} a_{i}$

The bosonic field operators obey the commutation relation

$\left[\phi(\mathbf{r}) , \phi(\mathbf{r'}) \right] = 0 \quad,\quad \left[\phi^\dagger(\mathbf{r}) , \phi^\dagger(\mathbf{r'}) \right] = 0 \quad,\quad \left[\phi(\mathbf{r}) , \phi^\dagger(\mathbf{r'}) \right] = \delta^3(\mathbf{r} - \mathbf{r'})$

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

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

$H = - \frac{\hbar^2}{2m} \sum_i \nabla_i^2 + \sum_{i < j} U(|\mathbf{r}_i - \mathbf{r}_j|)$

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

$H = - \frac{\hbar^2}{2m} \int d^3\!r \; \phi(\mathbf{r})^\dagger \nabla^2 \phi(\mathbf{r}) + \int\!d^3\!r \int\!d^3\!r' \; \phi(\mathbf{r})^\dagger \phi(\mathbf{r}')^\dagger U(|\mathbf{r} - \mathbf{r}'|) \phi(\mathbf{r'}) \phi(\mathbf{r})$

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, so a quantum field theory must be formulated right from the start. The process is known as "first quantization of a classical field equation."

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

$\left[ q_i , p_j \right] = \delta_{ij}$

For an electromagnetic field, the analogue of the coordinate variables are the values of the electrical potential φ(x) and the vector potential A(x) at every point x. This is an uncountable set of variables, because 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.

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.

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).

$a_2 \| N_1, N_2, N_3, \cdots \rangle \equiv$	$\sqrt{N_2}$	$\|N_1, (N_2 - 1), N_3, \cdots \rangle$
$a_2^\dagger \| N_1, N_2, N_3, \cdots \rangle \equiv$	$\sqrt{N_2 + 1}$	$\| N_1, (N_2 + 1), N_3, \cdots \rangle$

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