Content deleted Content added
Citation bot (talk | contribs) Removed proxy/dead URL that duplicated identifier. | Use this bot. Report bugs. | Suggested by Whoop whoop pull up | #UCB_webform 188/295 |
m →Quantum field theory: wikilink |
||
Line 400:
:<math>[\varphi(x), \partial_t \varphi(y)] = i \delta^3(x - y)</math>
for two simultaneous spatial positions {{mvar|x}} and {{mvar|y}}, and this is not a relativistically invariant concept. The results of a calculation ''are'' covariant, but the symmetry is not apparent in intermediate stages. If naive field-theory calculations did not produce infinite answers in the [[continuum limit]], this would not have been such a big problem – it would just have been a bad choice of coordinates. But the lack of symmetry means that the infinite quantities must be cut off, and the bad coordinates make it nearly impossible to cut off the theory without spoiling the symmetry. This makes it difficult to extract the physical predictions, which require a [[renormalization|careful limiting procedure]].
The problem of lost symmetry also appears in classical mechanics, where the Hamiltonian formulation also superficially singles out time. The Lagrangian formulation makes the relativistic invariance apparent. In the same way, the path integral is manifestly relativistic. It reproduces the Schrödinger equation, the Heisenberg equations of motion, and the canonical commutation relations and shows that they are compatible with relativity. It extends the Heisenberg-type operator algebra to [[operator product expansion|operator product rules]], which are new relations difficult to see in the old formalism.
| |||