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Quantum Theory

Elementary Quantum Mechanics. by R W Gurney

By R W Gurney

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The fact that gauge conditions eliminate the gauge freedom implies that their solutions intersect each orbit of the gauge group in exactly one point. Generically, such conditions cannot be chosen globally on the whole space of the fields, but, since we are working with small fluctuations only, it is not a problem. Let us illustrate the method by using a pure Maxwell theory. The gauge modes here have the simple form, ξμ = ∂μ λ, where the gauge parameter λ is a sufficiently smooth function. For any potential Aμ there is a gauge parameter λ such that after the corresponding transformation the potential satisfies the so-called Lorentz condition ∇ μ Aμ = 0.

1 Relativistic Inner Product 31 at most, one can easily show, that ∂L(f1∗ , f2 ) ∂L(f1∗ , f2 ) j μ (f1 , f2 )α = δα f 2 + δα f1∗ ∗ ∂f2,μ ∂f1,μ or ∂L[f1∗ , f2 ] ∂L[f1∗ , f2 ] ∗ f2 − f1 . 8) These statements are a particular case of what is known as the Noether theorem for global symmetries. We shall also deal with the Noether theorem for local symmetries in Chap. 8. The current j μ (f1 , f2 ) is called the Noether current. 9) which is called the relativistic inner product. The product is linear in the second argument f2 and anti-linear in f1 , and is Hermitian, f1 , f2 = f2 , f1 ∗ .

1) where for integer spin fields P (∂t , ∂k ) is a second order hyperbolic type partial differential operator. 74). 1). We are going to introduce a socalled relativistic inner product, f1 , f2 , between these solutions. The product is constructed through a conserved current corresponding to some global symmetry. Since the inner product and the current must depend on two fields instead of one, we have to double the number of fields in the quadratic form of the action. This is D. Fursaev, D. V. 2011 29 30 2 Quantum Fields done in the following way.

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