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A classical field theory is a physical theory that predicts how one or more fields in physics interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum field theories.
The topic broadly splits into equations of classical field theory and quantum field theory. Classical field equations describe many physical properties like temperature of a substance, velocity of a fluid, stresses in an elastic material, electric and magnetic fields from a current, etc. [1] They also describe the fundamental forces of nature ...
In mathematical physics, covariant classical field theory represents classical fields by sections of fiber bundles, and their dynamics is phrased in the context of a finite-dimensional space of fields. Nowadays, it is well known that [citation needed] jet bundles and the variational bicomplex are the correct domain for such a description.
For a scalar field theory with D spacetime dimensions, the only dimensionless parameter g n satisfies n = 2D ⁄ (D − 2). For example, in D = 4, only g 4 is classically dimensionless, and so the only classically scale-invariant scalar field theory in D = 4 is the massless φ 4 theory.
In fact, in quantum field theory physically interesting measures are concentrated on distributional configurations. That physically interesting measures are concentrated on distributional fields is the reason why in quantum theory fields arise as operator-valued distributions. [2] The example of a scalar field can be found in the references [3] [4]
When the canonical quantization procedure is applied to a field, such as the electromagnetic field, the classical field variables become quantum operators. Thus, the normal modes comprising the amplitude of the field are simple oscillators, each of which is quantized in standard first quantization, above, without ambiguity.
These parameters are represented, in the quantum field theory that approximates the string theory at low energies, by the vacuum expectation values of massless scalar fields, making contact with the usage described above. In string theory, the term "moduli space" is often used specifically to refer to the space of all possible string backgrounds.
The series commenced with What You Need to Know (above) reissued under the title Classical Mechanics: The Theoretical Minimum. The series presently stands at four books (as of early 2023) covering the first four of six core courses devoted to: classical mechanics , quantum mechanics , special relativity and classical field theory , general ...