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In/Equivalence of Klein-Gordon and Dirac Equation
| Content Provider | Semantic Scholar |
|---|---|
| Author | Hüttenbach, Hans Detlef |
| Copyright Year | 2014 |
| Abstract | It will be proven that Klein-Gordon and Dirac equation, when defined on an F-space of distributions, have the same set of solutions, which makes the two equations equivalent on that vector space of distributions. Some consequences of this for quantum field theory are shortly discussed. 1. Klein-Gordon And Dirac Equation I assume ~ ≡ 1 and c ≡ 1 throughout, denote with x = (x0, . . . , x3) ∈ R4 a point in space-time, where x0 is the time coordinate, and base the Minkowsi metric tensor g on the signature (+,−,−,−). With this convention, the KleinGordon is: Ψ := (∂2 0 − ∂2 1 − ∂2 2 − ∂2 3)Ψ = −m2Ψ, (1.1) and the Dirac equation is given as a matrix equation by (iγ0∂0 − · · · − iγ3∂3)Ψ = mΨ, (1.2) where γ0, . . . γ3 are anticommuting 4 × 4-matrices satisfying γ2 0 = −γ2 1 = −γ2 2 = −γ2 3 = 14, with 14 denoting the 4×4-unit matrix. The Dirac matrices are a possible representation of these matrices. (As a reference I refer to [4], [9], [2], [3], or any good book on quantum field theory.) 2. Definition of the F-space The operator iγ0∂0−· · ·− iγ3∂3 is called Dirac-operator. Rather than to discuss it in terms of Hilbert spaces, where it is unbounded and non-selfadjoint, I prefer a super space X, say, on which this operator is a linear, continuous mapping. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://vixra.org/pdf/1410.0154v1.pdf |
| Alternate Webpage(s) | http://vixra.org/pdf/1410.0154v2.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
