Quantum Field Theory

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. This problem was first examined by Max Born and Pascual Jordan in 1925 . 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 . In order to quantize this theory, they used the Canonical Quantization procedure. In 1927 , Paul Dirac gave the first consistent treatment of this problem. Quantum field theory followed unavoidably from a quantum treatment of the only known classical field, ie, Electromagnetism . The theory was required by the need to treat a situation where the ''number of particles changes''. Here, one atom in the initial state becomes an atom and a Photon in the final state.

It was obvious from the beginning that the quantum treatment of the electromagnetic field required a proper treatment of relativity. Jordan and Wolfgang Pauli showed in 1928 that Commutators of the field were actually Lorentz Invariant . By 1933 , Niels Bohr and Leon Rosenfeld had related these commutation relations to a limitation on the ability to measure fields at Space-like separation. 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. This thread of development was incorporated into Many-body Theory , and strongly influenced Condensed Matter Physics and Nuclear Physics .

What QFT is

Just as quantum mechanics deals with Operator s 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. Intuitively, this means that operators can be localized. This definition applies even to the cases of theories which aren't quantizations, and as such, is pretty general.

This is sometimes stated as "position is an operator in QM but is a parameter in QFT" but this statement, while accurate, can be very misleading. 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 Quasiparticle s) 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

:

\hat{\mathbf{x}}(t)

and

\hat{\mathbf{p}}(t)

,

we now look at operators generated by

:

\hat{\phi}(\mathbf{x},t)

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 ). Only the Heisenberg picture is manifestly Lorentz Covariant .

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...) This lowest energy eigenstate is called the vacuum in particle physics and the ground state in condensed matter physics. (QFT appears in the continuum limit of condensed matter systems)

This simple explanation of what QFT really is, is often obscured in treatments which jump straight to the Path Integral approach, which is a good computational technique but often obscures the underlying ideas.

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

	:<math> A 2 N 1, N 2, N 3, \cdots Angle	\sqrt{N_2} \mid N_1, (N_2 - 1), N_3, \cdots angle </math>
	:<math> A 2^\dagger N 1, N 2, N 3, \cdots Angle	\sqrt{N_2 + 1} \mid N_1, (N_2 + 1), N_3, \cdots angle </math>
	:<math>a K^\dagger\,a K\cdots, N K, \cdots Angle	N_k \cdots, N_k, \cdots angle</math>
	:<math> C J N 1, N 2, \cdots, N J	0, \cdots angle = 0 </math>
	:<math> 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 </math>
	:<math> 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 </math>
	:<math> C J^\dagger N 1, N 2, \cdots, N J	1, \cdots angle = 0 </math>

The bosonic field operators obey the commutation relation

:

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

where

\delta(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 = - rac{\hbar^2}{2m} \sum_i

::<math>\psi(t) Angle

e^{iH_It} \psi(0) angle = \left[1+iH_It-rac12 H_I^2t^2 -rac i{3!}H_I^3t^3 + rac1{4!}H_I^4t^4 + \cdots ight] \psi(0) angle</math>

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Quantum Field Theory