Spin, Lorentz Invariant Equation for Free Probability and Lorentz Invariant Physical Equation

Spin, Lorentz Invariant Equation for Free Probability and Lorentz Invariant Physical Equation Francesco R. Ruggeri Hanwell, N.B. Oct. 11, 2025 In this note, we suggest that a fundamental idea of free particle quantum mechanics is the wavefunction exp(-iEt+i p dot r) which we call a complex probability. This probability, however, does not describe the full physics of a free object. For example, it does not tell one if the object has rest mass or not, or whether it has electric and magnetic fields as in the case of a photon. Thus, we suggest that there must exist a Lorentz scalar equation describing the full physics of the problem, but that this equation makes use of the probability exp(-iEt+ip dot r), but not necessarily in a linear form (as in the continuity equation of energy density and Poynting momentum for a photon). At the same time exp(-iEt+ip dot r) is an eigenfunction of linear operators i d/dt partial and -i grad partial. Thus, we postulate that at the root of any physical Lorentz invariant equation using functions based on exp(-iEt+i p dot r), there should be a corresponding Lorentz scalar equation in exp(-iEt+i p dot r). Such an equation must be based on the operators id/dt partial and -i grad partial, but as a Lorentz scalar, it must contain an operator which forms a dot product with - i grad. If px, py, pz and E are the pieces of information of interest, then this new vector is not based on x-space etc, but may be a set of matrices which allow this invariant base equation to become the Lorentz scalar physical equation. In other words, one needs to add a kind of geometry to exp(-iEt+ip dot r) to allow it to describe the full physics of a problem. This geometry seems to be hidden in the way the E, px, py and pz variables are interrelated. We consider both the spin ½ and spin 1 photon cases. Quantum Free Particle and exp(-iEt + i p dot r) In previous notes, we argued that given a collision of E1,E2 and p1,p2 with conservation of both, one does not know the outcome values Ei, Ek and pk,pl (one dimension). If one assumes equal probability for any pair which maintains conservation, then one would expect a probability of the form: exp(i E) and exp(i p) one dimension ((1)) “I” is used because the probability of a free particle cannot be real, i.e. one does have any weight as in an ideal gas. exp(ip) and exp(-ip), however, must have the same value and so ((1)) cannot be correct. One must combine p with a variable which also changes when one changes the direction of the x axis. This suggests then: exp(-iEt) and exp(ipx) to have an overall Lorentz invariant scalar probability ((2)) A fundamental issue now arises. The probability exp(-iEt+i p dot r) does not describe the full properties of a free object. For example, exp(-iEt+i p dot r) could apply to a particle with rest mass or a photon with no rest mass. A particle with rest mass is described by the Lorentz invariant scalar equation: -EE = cc p dot p + momo cccc ((3)) A photon is described by both: E = c |p| ((4)) as well as an equation using electric and magnetic fields, i.e. d/dt partial( .5eo E dot E + .5/uo B dot B) + 1/uo grad dot E x B = 0 ((5)) ((5)) depends one exp(-iEt+i p dot r) through E = C1 Real (exp(-iEt+i p dot r) vector along r and B = C2 Real (exp(-iEt+i p dot r) vector along r ((6)) ((3)) is nonlinear in the operators id/dt partial and -i grad partial, but linear acting on exp(-iEt+ip dot r), whereas ((5)) is linear in the operators id/dt and -igrad, but quadratic in the probability exp(-iEt+i p dot r). We postulate the following. We argue that the free object probability exp(-iEt+i p dot r) is a key underlying partial description of a free object. It is an eigenfunction of id/dt partial and -i grad partial, but is missing information contained in equations such as ((3)) and ((5)). We suggest that just as ((3)) and ((5)) are Lorentz invariant equations which make use of the E, px,py,pz, there must exist a Lorentz scalar equation, linear in id/dt partial, -i grad partial for exp(-iEt+i p dot r). Furthermore, this equation for the probability must be the foundation upon which one may create the more general physical equations such as ((3)) or ((5)). The two equations cannot be independent. This suggests that one must find a vector to use in a dot product with p (or -i grad). This vector, however, does not seem to be the r vector or any other physical vector. Instead, it seems that it is required to describe a kind of hidden geometry in the problem. To see this specifically, we consider an abstract math vector consisting of matrices. In such a case, one should have: -mo Matrix A + Matrix0 (id/dt partial) + Sum i=1,3 Matrix-i (id/dx) } exp(-iEt+ipx) object = 0 ((7)) ((7)) must be a building block equation for ((3)), with the obvious operation being the multiplication of both sides of ((7)). This yields various conditions for the matrices. Dirac has already solved this problem and found that the: Matrix-i (i=1,2,3) are 4x4 matrices with the Pauli 2x2 spin matrices on the anti-diagonal, with the second one appearing with a minus sign. ((8)) The point is that we suggest that ((7)) should exist a priori because there must be a Lorentz scalar equation in id/dt and -igrad acting on exp(-iEt+ip dot r) and ((3)) cannot be independent of this equation. If this is the case, then one should be able to use ((7)) with mo=0 for a photon.The same basic building block idea should hold. In a previous note, we have shown that one may take ((5)) and linearize it by factoring out Electric field + i B field to obtain: d/dt (El+iB) + eijk d/dxk (El+iB) ((9)) Here eijk is the Levi-Civita symbol and ij are the matrix entries, while k=1,2,3 describes the matrix itself. As a result, the geometry is different in ((9)) because the physical Lorentz scalar equation ((5)) differs from ((3)). In other words, spin describes in a geometrical manner how a basic building block equation linear in id/dt and -i grad acting on exp(-iEt+ip dot r) with a vector or matrix M-i dotted with d/dxi may be used to create the Lorentz scalar physical equation as the two cannot be independent. Conclusion In conclusion, we try to understand how spin arises. We suggest that without considering any quantum mechanical free particle features, there exist physical equations such as ((3)) and ((5)) for a particle with rest mass and a photon described in terms of electric and magnetic fields. These are Lorentz invariant equations. We next suggest that the quantum mechanical feature of a free object follows from postulating that a given E1,E2, p1, p2 (one dimension) can create any Ei, Ej, pl, pk outcome pairs with equal probability as long as energy and momentum are conserved. We then argue that the probability must be described by exp(-iEt+i p dot r) which is a Lorentz invariant. Thus, the physical Lorentz scalar equations ((3)) and ((5)) actually depend on exp(-iEt+i p dot r), but this does not mean that the operators id/dt and -igrad appear linearly (they do in ((5)) but not in ((3))). Nor does it mean that exp(-iEt+i p dot r) appears linearly (it does in ((3)), but not in ((5))). We suggest that exp(-iEt+i p dot r) is an eigenfunction of -id/dt partial and i grad partial and so there should exist a Lorentz scalar equation for exp(-iEti p dot r). We further suggest that the vector -i grad should form a dot product with a math vector of matrices. This Lorentz scalar equation for exp(-iEt+i p dot r) would then serve as the building block for the physical Lorentz scalar equation, as the two cannot be independent. Thus, the vector of matrices determines what physical Lorentz scalar equation results from a building block exp(-iEt+i p dot r) with matrices equation. In the case of ((3)), the matrices linked with p are 4x4 one with 2x2 Pauli matrices along the diagonal with the second matrix having a factor of -1. In the case of ((5)), eijk is the matrix with i,j representing the matrix elements. In other words, the spin form describes a hidden geometry required to convert the building block Lorentz scalar equation for exp(-iEt+i p dot r) into the physical Lorentz scalar equation.