First Principles

Particles from First Principles

Introduction

The idea behind the following is to put the discrete indices on the same footing as the continuous variables. With nothing more complicated than momentum conservation and charge conservation we shall attempt to deduce the Standard Model.

Definitions

Let define a field Φ(k,N) where k is a 4-vector of continuous values and N is a vector of discrete values of unknown (or even infinite) length. k shall be called the momentum vector and N shall be called the charge vector. The vectors N could, for example, be vertices on a polytope or lattice with the property that each vector is the sum of at least one pair of other vectors.

Neutral Fields

Now we must also define some more (vector) functions at the special points where k=0 or N=0. Where there is no momentum of no-charge.

The neutral fields are given by the Z functions, which can be contracted with a charge:

Z(k,0).N

The sigma functions are given by the functions, which can be contracted with the momentum:

σ(0,N).k

>Explain why we must have neutral fields to close the group.
>Should there be neutral currents for spin?

The sigma are discrete functions of the charge. They don't change with momentum. If we contract them by summing over opposite charges we get a quadratic function in k. The only quadratic function in k that we are allowed (because we want Lorenz symmetry) is the square of the length of the vector of k plus, optionally, a mass squared term. So we must set them equal to values such that:

Σ σ(0,N).k σ(0,-N).k  = k.k

summed over the charges N. One way to do this is to set:

σ(0,[+1,+1,0,..]).k = k1+k0

σ(0,[+1,-1,0,..]).k = k2+ik3

σ(0,[-1,+1,0,..]).k = k2-ik3

σ(0,[-1,-1,0,..]).k = k1-k0

For all vectors A and B. Another representation would be [±2,0,0,..] and [0,±2,0,..]. These are simply the Pauli/Dirac matrices for chiral fields. (Which components we choose effects the interpretation of those components). Therefore mapping the momentum components to lattice vertices means we can treat spin as just another kind of charge. One that just happens to coincide with kinetic components. 

Conservation of Momentum and Charge

In any interaction the sum of any the components of momentum or charge for the fields is always zero. For example we can have terms:

Φ(k,N) Φ(k',N') Φ(-k-k',-N-N') 

Φ(k,N) σ(0,-N).k Φ(-k,-N-N') 

Φ(k,N) Z(k',0).N Φ(-k-k',-N)

Φ(k,N) σ(0,N').k σ(0,N'').k Φ(-k,-N-N'-N'') 

Note: The momentum that is contracted with the sigma or the charge that is contracted with the Z fields is not involved in the conservation principle.

The first term could be a gluon interacting with two quarks. The second the kinetic term for an electron. The third a Z particle interacting with a neutrino. The fourth is Maxwell's equations.

Names of Charges

Let N = [J1, J2, R1, R2 ,... ,R8,...]

The spin of a particle is given by |J|^2/2. Thus a fermion may have J=[1,0] or [0,-1] for example. The reason is because we have used these components for the sigma functions.

Φ(k,[+1,+1,...]) = A1(k)+A0(k) (spin 1)

Φ(k,[+1,0,...]) = ψ1(k)  (spin 1/2)

Φ(k,[-1,0,...]) = ψ2(k)

Φ(k,[0,+1,...]) = ψ*1(k)

Φ(k,[0,-1,...]) = ψ*2(k)

Φ(k,[0,0,...]) = Φ(k)   (spin 0)

This can be put in a matrix:

[A1+A0     ψ1      A2+iA3   ]
[ψ1*           Φ           ψ2           ]
[A2-iA3     ψ2*     A1-A0    ]

These can be components of the group O(5) (=Sp(2)). Thus the Dirac/Pauli matrices are just the spin-1 part of the O(5) group.

The electric charge of a particles is given by (R1+R2+R3-3*R4)/6.

For E8 charges permutations of R=(±1,±1,±1,±1,±1,±1,±1,±1) or (±2,±2,0,0,0,0,0,0)

Higgs and Mass
The Higgs particles are spin 0 particles of the form (to give fermions mass) and mixing angles:

R=(0,0,0,0,±2,±2,0,0)

R=(0,0,0,0,±2,0,±2,0)

R=(0,0,0,0,±2,0,0,±2)

and also neutral particles contracted with (to give Z, W± masses but not A)

R=(1,1,1,-3,0,0,0,0)

and...?

Supersymmetry

Extra components of the charge encode supersymmetry?? (Super-charges are NOT conserved in an interaction!)

N = [...,S1,S2,S3,...S8]

In this example of N=8 supersymmetry the spin is given by (S1+S2...+S8)/4.

Spin 2? Gravity?

Diffeomorphism
This can be written down easily for the phi function:

d Φ(x,N)  = h(x).∂ Φ(x,N) + Σ σ(x,N).∂ σ(x,N').h(x) Φ(x,N-N'-N'')

Summed over N' and N''. When h(x) = M.x+P this is Poincaré invariance (Lorenz+translational).

Lattice Points?

The question is what lattice are the points on? How can we find that out? Could it be the Leech Lattice? (Which contains E8xE8).

Note these give approximately the right number of particles but the wrong charges:

g=(±1,±1,±1,±1 | 0,0,0,0)   (1)

(8)

A=(±1,±1,0,0| ±1,±1,0,0)   (144)

ψ=(±1,0,0,0 | ±1,±1,±1,0)   (64)

Φ=(0,0,0,0 | ±1,±1,±1,±1)  (16)

70x16=1120 components altogether.

(Could this be from an alternative E8 rep?)

g=(±1,±1,±1,±1 | 0,0,0,0,0,0,0,0)

A=(±1,±1,0,0| 1,1,0,0,0,0,0,0)  + (±1,±1,0,0| 0,0,0,0|E8xE8)        

ψ=(±1,0,0,0 | 1,1,1,0,0,0,0,0) [12]  + (±1,0,0,0 | 1,0,0,0|E8xE8)  [24]

(J1,J2,J3,J4|N1,N2,N3,N4|R1,...R8,r1,..r8) [24] Leech Lattice!

Guage Invariance
(Combined Super-Diffeomorphism and Gauge-Invariance?)
dΦ =<hxΦ>  = (h.dΦ - Φ.dh) + (dh + hΦ) + a Φ(d.h) 

Super-Diffeomorhpism
dΦ =(h.d)Φ - (Φ.d)h  + a Φ(d.h) 
Guage invariance:
dΦ =dh + Φh =D[Φ]h

Pre-geometric:
dΦ(p) = h(k)h(q)Φ(p-j-q) + h(k)Φ(p-k) ???

h(k)Φ(p-k) --> h(x)Φ(x) + DΦ(x) - Dh(x)

Gauge Variables
Λ(x)=
[h1+h0     ψ1      h2+ih3   ]
[ψ1*          λ           ψ2        ]
[h2-ih3     ψ2*     h1-h0    ]

Gauge group is a Group Multiplication of Fields
<Φ1xΦ2> = Φ3
L[f]L[g]=L[fDg-gDf+Df+fg]

Φ(k)Φ(p)Φ(k-p) + Φ(k)Φ(0)Φ(-k)