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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...
7.7 is obtained by integrating 9.1 over one time interval. Thus, in the restriction to integer values, 9.1 is identical to 7.7, the difference equation for a non-interacting particle. It is therefore an expression of the same relationship or law. As an equation of the wave function, the right hand side of 9.1 is a scalar, whereas the left hand side is the time component of a vector whose space component is zero. So 9.1 is not manifestly covariant. A covariant equation which reduces to 9.1 requires the scalar product between the vector derivative, ∂, and the wave function and has the form, for some vector operator Γ i∂ ⋅ Γf = m f 9.2 Then the time evolution of the position function in any reference frame Ν is the restriction of the solution of 9.2 to Ν at time t ∈ Τ . As discovered by Dirac [4], there is no invariant equation in the form of 9.2 for scalar f and the theory breaks down. To rectify the problem a spin index is added to Ν ΝS = Ν ⊗ S for v ∈ 0 where S is a finite set of indices. When there is no ambiguity we write Ν = Ν S , and the constructions of the vector spaces, *, * and (, go through as before. When we wish to make the spin index explicit we write | x 〉 = | x , α 〉 = | x 〉 α normalised by 2.1 ∀( x, α ), ( y, β ) ∈ Ν S 〈 x , α| y , β 〉 = 〈 x | y 〉 α β = δ x y δ α β 9.3...
Particular interactions can be postulated as operators with the general form of 13.11, we can examine whether the resulting theoretical properties correspond to the observed behaviour of matter...
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...
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)...
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]...
Yang-Mills Fields and Quark Confinement
Following the arguments of section 9 we seek to replace the discrete time evolution equation 7.2 with a covariant equation of motion, which should have the form of 9.2, but with the addition of an interaction term (i∂ ⋅ Γ + H(x))f = m f 19.1
Now we seek to generalise to the case where we do not have a simple Dirac particle, but a composition of Dirac particles |x i, q i µ i〉 where i is an index (colour), and q i is a flavour of quark...
This expression for the field operator is formally like that used when we consider colour compositions of singlet quarks, and to that treated in the literature, e.g. [15](15.19). We draw attention to this because, although formally similar, the symmetry states are interpreted a little differently from the standard model where fields are considered separately from the Fock space on which they act. Historically SU(2), or isospin, was introduced by considering a nucleon as a particle which could be either proton or neutron. But the symmetry in 19.4 was found by considering a particle which is a composition of colours. In other words 19.4 represents the field operator for a particle containing three colours in symmetrical combinations, not one of any colour with symmetrical probability amplitude. Following the construction of qed we seek to construct an interaction operators H ( x ) in which a vector boson is emitted or absorbed by the baryon. The treatment of the electromagnetic interaction follows that for leptons, but the Dirac adjoint is
0 ˆ ψ i µ ( x ) = ψ† i α ( x ) γ α µ...
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....
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