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It is straightforward to see that 7.2 is in some sense approximated by the time evolution equation | f 〉 t + 1 = µ e –i I ( t ) | f 〉 t found in, for example, lattice field theory [13]. Lattice field theory does not describe the model of simple particle interactions considered here, so there is motivation for a somewhat modified treatment. Observe that Hilbert space is information space at some time t, ( = ( ( t ) , so the interaction is...
7.6 has a straightforward solution with µ 2 = 1 . and I = I† . Although strictly non-linearity implies that I is not hermitian, I 2 ( t ) does not appear in the physical model, as discussed above, and we may treat I as hermitian and will refer to it as such. 7.2 can be interpreted literally as meaning that in each instant particle either interacts or does not interact. In the latter case the state remains the same and is multiplied by a phase, µ = e – i E , E ∈ 4 so that 7.2 reduces to | f 〉t + 1 = e –i E | f 〉 t 7.7 is a geometric progression with solution | f 〉t = e –i E t | f 〉0 7.8 7.7...
The Photon Field
Photons are bosons, and having zero mass, the photon is its own antiparticle so that | x , α〉 = | x , α〉 ...
The Dirac Field
ψ α ( x ) = |x, α〉 + 〈x, α| 16.1
Definition: By 13.8, the Dirac field is
We know from observation that a Dirac particle can be an eigenstate of position...
Position kets are a basis, so 16.2 reduces to Z( x ) = | x 〉〈 x | up to the resolution of the apparatus...
The Non-Perturbative Solution
Because local phase is a freedom in the definition of Hilbert space and can give no physical results we have that U(1) local gauge symmetry is preserved in interaction (section 2)...
Feynman Rules
˜ ˜ ˜ ˜ Definition: For any vector p, such that p 2 = m 2 , let p = ( p 0, p ) be a matrix for any p 0 ∈ 4 ...
Discrete Quantum Electrodynamics
Definition: The step function is given ∀x ∈ 4 , by Θ(x) = 0 1 if x ≤ 0 if x > 0
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Let | g 〉 ∈ ) be a measured state at time T...
Indeed our analysis of the origin of the ultraviolet divergence is essentially the same as that given by Scharf [17]...
By interchanging ( x n, α ) and ( x j, β ) in the diagram, we find for the adjoint propagator arrowed from j to n
T ˆ ˆ j n n j Θ ( x 0 – x 0 ) 〈x j, β|x n, α〉 + Θ ( x 0 – x 0 ) 〈 x n, α |x j, β〉
18.14
18.14 is identical to 18.12, the expression for the Dirac propagator arrowed from j to n, so we do not distinguish whether an arrowed line in a diagram is generated by the field or the adjoint field...
There are two straight-
Discrete Quantum Electrodynamics
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forward ways of forming states which are symmetrical in the colour index...
Acknowledgements
I should like to thank a number of physicists who have discussed the content and ideas of this paper on usenet, particularly Paul Colby, Matthew Nobes, Michael Weiss and Toddlius Desiato for their constructive criticism of earlier versions of the paper, John Baez for instruction and advice about the current status of field theory, and the moderators of sci.physics.research (John Baez, Matt McIrvin, Ted Bunn & Philip Helbig) for their vigilance in pointing out lack of clarity in expression in describing the model...
[16] Rescher N.: Many Valued Logic, McGraw-Hill, New York, (1969) [17] Scharf G.: Finite Quantum Electrodynamics, Springer, Berlin (1989) [18] Streater, R...
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