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## Quantum field theory
## Shortcomings of ordinary quantum mechanicsQuantum field theory corrects several deficiencies of ordinary quantum mechanics, which we will briefly discuss. The Schrödinger equation, in its most commonly-encountered form, is
m its mass, and V an applied potential energy.
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
The second problem occurs when we seek to extend the equation to large numbers of particles. It was discovered that quantum mechanical particles of the same species are indistinguishable, in the sense that the wavefunction of the entire system must be symmetric (bosons) or antisymmetric (fermions) when the coordinates of its constituent particles are exchanged. This makes the wavefunction of systems of many particles extremely complicated. For example, the general wavefunction of a system of
r are the coordinates of the _{i}i-th particle, φ are the single-particle wavefunctions, and the sum is taken over all possible permutations of _{i}p elements. In general, this is a sum of N! (N factorial) distinct terms, which quickly becomes unmanageable as N increases.## Quantum fields
Both of the above problems are resolved by moving our attention from a set of indestructible particles to a
We should mention two possible points of confusion. Firstly, the aforementioned "field" and "particle" descriptions do
In second quantization, we make use of particle indistinguishability by specifying multi-particle wavefunctions in terms of single-particle
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 a has the following effects:_{2}
Finally, we introduce
E is the kinetic energy of the _{k}k-th momentum mode. In fact, this Hamiltonian is useful for describing non-interacting phonons.## Wightman axiomsThis is one of the many attempts to put quantum field theory on a firm mathematical footing.## Suggested readingPeskin, M. and D. Schroeder. 1995. An Introduction to quantum field theory. | |||||||

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