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Spin, Lorentz Invariant Equation for Free Probability and Lorentz Invariant Physical Equation Part 2 Francesco R. Ruggeri Hanwell, N.B. Oct. 13, 2025 In Part 1, we argued that the quantum free particle form exp(-iEt+ i p dot r), which holds for both a photon and particle with rest mass, follows from conservation of energy and momentum, an idea known in classical physics. This conservation, however, would only require the forms exp(-iEt) and exp(ip). This, however, has two problems. The probability is not Lorentz invariant and exp(ip) is not equal to exp(-ip). Both of these problems may be solved by using exp(-iEt+ip dot r), but this now involves a new idea, namely that of invariance in space and in time. One has not only the idea of conservation of E and p, but the notion of invariance in space and time because one may obtain E and p values using id/dt partial and -i grad partial (hbar=1). Thus, we argue that in free particle quantum mechanics, one must be sensitive to invariances in space-time linked to constraints. In other words, one need not only consider conservation, but any physical equation. In Part 1, we postulated that since exp(-iEt+i p dot r) is an eigenfunction of id/dt partial and -i grad partial (hbar=1), one should be able to write a linear equation in these operators representing a Lorentz scalar equation. This linear equation serves as building block for physical Lorentz scalar equations such as the Klein-Gordon equation and d/dt (.5eo El dot El + .5/uo B dot B) + 1/uo grad dot El x B = 0, where El and B are the electric and magnetic fields of the photon. Here, we note that exp(-iEt+i p dot r) arose from two considerations, conservation of E and p (which is a constraint) and invariance in space and time. We note that both the Klein-Gordon and Poynting continuity equations are constraint equations and involve various invariances and geometry conditions. For instance, in the Klein-Gordon equation, one has p dot p which is rotationally invariant and in the Poynting continuity equation El x B indicates orthogonality of two vectors which is also linked to the idea that one can rotate the two perpendicular vectors as long as they remain perpendicular. Thus, we suggest that given the notion of constraint equations with invariances/geometrical features, a linear equation for exp(-iEt+i p dot r) using id/dt and -igrad may involve operators linked to this invariance. In particular, one may be forced to dot a set of matrices with -i grad and that this gives rise to spin. exp(-iEt+ i p dot r), Conservation and Invariance In Part 1, we noted that the form exp(-iEt+i p dot r) is linked to the classical ideas of conservation of energy and momentum for a free particle. The probability of a free particle is a unit modulus complex number because there is no real weight unlike in an ideal gas. Conservation, however, would lead to: exp(iE) and exp(ip) ((1))
Such a probability is not Lorentz invariant and exp(ip) does not equal exp(-ip) which is a problem. These issues may be solved using: exp(-iEt + i p dot r) ((2)) The point we make is that ((2)) is now not simply associated with conservation of E and p, but also with invariance in space and time because ((2)) is an eigenfunction of: id/dt partial and -i grad partial ((3)) d/dt and grad are generators of translations in time and space. Thus, it seems like there are two interconnected ideas: conservation and space-time invariances involved in free particle quantum mechanics. -Et + p dot r is a Lorentz scalar which is invariant as well. We suggest that these ideas may be extended to spin, but to do that one must identify conservation (or a constraint because conservation is a type of constraint) and invariance-geometry. Spin From Invariance Considerations In Part 1, we suggested that one create a Lorentz scalar equation linear in id/dt and -grad which may be used as a building block for creating relevant physical Lorentz scalar equations which exist. In particular, this involved writing: (sx) px + (sy)py + (sz) pz where si is a matrix ((4)) The relevant physical equations we used in Part I are: -EE + cc p dot p = -momocccc ((5a)) and d/dt (.5eo El dot El + .5/uo B dot B) + 1/uo grad dot (ElxB) = 0 ((5b)) We suggest here that these are constraint equations (because that is the nature of an equation).There is also a certain kind of conservation involved as E,p are matched with mo and ((5b)) is a continuity equation which represents conservation directly. We also suggest that there is invariance-geometry in both of these equations. In particular, p dot p in ((5a)) is invariant under rotation and ElxB represents perpendicular El and B pieces and is invariant under rotation. These are the ingredients which led to the operators id/dt partial and -grad partial for exp(-iEt+i p dot r) and so we suggest that there is no reason not to expect some operators linked to these invariances in a linear equation involving id/dt, -i grad and exp(-iEt+ip dot r), in particular matrix operators. In Part I we discussed that one may write: Mo Matrix 1 + E Matrix 2 + (sx) px + (sy)py + (sz) pz = 0 ((6))
For mo not=0, we noted in Part 1 that Dirac found 4x4 matrices, with the sx ones consisting of the 2x2 Pauli matrices on the off-diagonals with the second one carrying a minus sign. This leads to the notion of spin ½ for a particle with rest mass. In the case of mo=0, one has a photon equation. We wish to examine this in more detail. If one writes: EE = p dot p cc ((7)) one may argue that one has both a constraint equation and rotational invariance in p dot p. As a consequence, one may write: E I = sx px + sy py + sz pz c ((8)) (I is the identity matrix) If only sz is nonzero, then: E I = sz pz c ((8)) The problem with ((8)) is that one may also write: E = pz c ((9)) ((9)) does not contain any matrix. As a result, even though one may write an equation ((8)) containing a matrix operator sz, it is not necessary to do so, i.e one is not constrained to do so. We thus argue that a photon does not have spin ½ because there is no restriction requiring the presence of a Pauli matrix. In the case of mo not=0, there is because given: EE = pzpz cc + momocccc ((10)) this cannot be written in linear form using only numbers. Dirac showed that matrices are required. Thus, the constraint and geometry-invariance require spin matrices in the linear equation for id/dt and -igrad. Given that a photon does not have spin ½, does that mean that there is no constraint equation with an invariance-geometry which is forced to be present in a linear equation containing id/dt and -i grad? We argued in Part 1, that the Poynting continuity equation ((5b))is exactly such an equation. In fact it is the Levi-Civita symbol eijk in: (El x B) i = eijk El-j B-k ((11)) which remains present in a linearized form of ((5b)) i.e. d/dt (E+iB) (i) + eijk d/dxj (E+iB) (k) = 0 ((12)) Thus, the geometry-rotational invariance in ((5b)) carries over into the linear equation in id/dt, -igrad and so spin is present, but based on eijk and so this is spin 1.
Conclusion In conclusion, we argue that the creation of exp(-iEt+i p dot r) (the quantum wavefunction or probability for a free particle) which holds for both a particle with rest mass mo and a photon, is based on two ideas. First, there is the notion of conservation of E and p, and secondly, there is Lorentz invariance of -Et+p dot r and exp(i p dot r) → exp(i p dot r) for p-> -p and r-> -r. This leads to idd/t and -grad which are generators of translation in time and space as well as exp(-iEt+i p dot r) being an eigenfunction of these operators. We suggest here that a conservation equation is really a constraint. We note in Part 1, that other physical Lorentz scalar constraint equations exist such as the Klein-Gordon equation (based on -EE = p dot p cc + momocccc) and the Poynting continuity equation ((5b)) which is directly linked to conservation. These equations also contain rotational invariance p dot p and El x B (which has the added feature or orthogonality). In Part 1, we showed that the Klein-Gordon equation leads to spin ½, i.e. is linked to Pauli matrices being present in a linear equation of matrices dotted with -id/dxi for mo not=0. One might argue that for mo=0, these matrices may still be present. We argue that although EE = cc pz pz (for px=py=0) may be written as: E I = c pz sz, it may also be written as E = c pz and so the presence of a matrix is not required. Thus a photon is not linked with spin ½ as a Pauli matrix is not required even though there is rotational invariance. Thus, the presence of invariance is not enough. It must force the presence of a matrix in the linear equation in id/dt and -grad. Even though EE = p dot p c does not force a matrix, the Poynting continuity equation does. We argue that spin arises when a physical Lorentz scalar equation (constraint) requires the presence of a matrix dotted with -i grad in a linear equation in id/dt, -igrad. This linear equation is based on exp(-iEt+i p dot r), the free particle probability.
