In two previous notes (Parts 1 and 2), we argued thta spin matrices arise when one attempts to separate a quadratic equation (continuity equation with the Poynting vector for a photon, Klein-Gordon equation for a spin .5 particle) into two factors. These appear as generalized numbers (i.e. matrices) which may be multiplied together without considering vector operations such as the cross or dot product. Here we try to examine the problem in an alternative way. We note first that in the case of the Klein-Gordon equation one has a 4-vector containing (p vector, E) or (r vector, t) to cite two examples. These transform under the unitary Lorentz transformation, i.e. via a unitary 4x4 matrix. A subset, however, namely p(x), p(y),p(z) or x,y,z of this 4-vector transforms physically under rotations in 3-space, i.e. via a 3x3 unitary matrix. Furthermore, there is an invariance under this 3x3 matrix transformation. In particular, objects such as the dot product or cross product do not change. Classically, one has no issue with vector operators in an equation. In quantum mechanics, however, a free particle is described by the wavefunction exp(-iEt+ip dot r) and the operator i d/dt partial and -i grad return E and p (vector). exp(-iEt+ip dot r), however, describes a translational invariance in space and time, i.e. there is a periodicity. This invariance is clearly seen if the complex conjugate of the free particle wavefunction is multiplied by the wavefunction to yield 1 everywhere.Thus one starts with a quadratic equation W*W and linearizes it to obtain the function W(x,t). The operators which act on W are the generators related to the invariance in W*W. For the quantum free particle, these are id/dt and -i grad.. We suggest that if such a scheme exists for this translational invariance in t and r, one should treat the rotational invariance in the Klein-Gordon equation in a similar manner. In other words, there should exist operators and a function associated with rotational invariance. In this case, the operator is a matrix (Pauli matrices) and the function a spinor.
In two previous notes (Parts 1 and 2), we argued thta spin matrices arise when one attempts to separate a quadratic equation (continuity equation with the Poynting vector for a photon, Klein-Gordon equation for a spin .5 particle) into two factors. These appear as generalized numbers (i.e. matrices) which may be multiplied together without considering vector operations such as the cross or dot product. Here we try to examine the problem in an alternative way. We note first that in the case of the Klein-Gordon equation one has a 4-vector containing (p vector, E) or (r vector, t) to cite two examples. These transform under the unitary Lorentz transformation, i.e. via a unitary 4x4 matrix. A subset, however, namely p(x), p(y),p(z) or x,y,z of this 4-vector transforms physically under rotations in 3-space, i.e. via a 3x3 unitary matrix. Furthermore, there is an invariance under this 3x3 matrix transformation. In particular, objects such as the dot product or cross product do not change. Classically, one has no issue with vector operators in an equation. In quantum mechanics, however, a free particle is described by the wavefunction exp(-iEt+ip dot r) and the operator i d/dt partial and -i grad return E and p (vector). exp(-iEt+ip dot r), however, describes a translational invariance in space and time, i.e. there is a periodicity. This invariance is clearly seen if the complex conjugate of the free particle wavefunction is multiplied by the wavefunction to yield 1 everywhere.Thus one starts with a quadratic equation WW and linearizes it to obtain the function W(x,t). The operators which act on W are the generators related to the invariance in WW. For the quantum free particle, these are id/dt and -i grad.. We suggest that if such a scheme exists for this translational invariance in t and r, one should treat the rotational invariance in the Klein-Gordon equation in a similar manner. In other words, there should exist operators and a function associated with rotational invariance. In this case, the operator is a matrix (Pauli matrices) and the function a spinor.
The photon spin matrix eijk (Levi-Civita symbol with i representing the matrix identifier and jk, its elements) follows from the continuity equation: d/dt (partial) {aEl dot El+bB dot B) + grad dot c El x B (where El and B are electric and magnetic field vectors). The strategy (as shown in (1)), seems to be to replace the vector operator x (cross product) with a matrix operator which acts on (El+iB) (vector). This matrix also has eigenvectors and so represents an operator. Thus -id/dx partial and id/dt are operators which act on a probability distribution exp(-iEt+ipx) which has physical implications and spin seems to be associated with an operator which also may act on a physical state. In other words, a physical state or degree of freedom gives rise to vector operator properties and does not simply pull out a number like id/dt partial yielding E for a free particle. We try to apply this idea to the Klein-Gordon equation -d/dt d/dt parital W(r,t) + grad dot grad W(r,t)= -momo W(r,t) (c=1). We argue that one wishes to replace the vector operation: dot product between grad and grad with an operator (matrix) which acts on a physical state (spinor). In other words, we argue that the vector operation of the dot product (like the cross product ) in the photon case, is caused by an operator (which acts on a physical state). We note that one has quadratic type equations, so the one linearizes, with the operator, i.e. matrix, appearing in each expression. Thus one obtains anti-commutation relationships when creating the quadratic form by multiplying the two linear expressions without using any vector operations. In other words, it is the presence of a physical state or degrees of freedom which automatically create the constraints introduced by the vector operator.
We have pointed out in previous notes that the free particle wavefunction exp(-iEt+ipx) holds for both a photon and particle with rest mass, while the latter has spin ½ and the former, spin 1. We thus suggested that spin is obtained in a very different manner from exp(-iEt+ipx). Here, we start with the Lorentz invariant -EE+ p dot p cc = -momocc ((1)) and argue, as in (1), that it must be linearized to obtain a linear equation in the intrinsic force, namely mo which is linked to inertia, because force acts in a linear form in nature (e.g. F=dp/dt). At this point, however, p is a vector and there is no concept of -ihbar grad. One may mathematically linearize ((1)) using the approach of Dirac (except that now p is a vector of numbers not an operator). This leads to 4x4 matrices with off diagonal Pauli matrices (multiplied by 1 and -1). One may see that the rotational invariant p dot p may be linearized to s dot p (s = three 2x2 Pauli matrices) because these matrices anticommute. Thus, a 2x2 Pauli matrix is associated with rotational invariance. At this point, this Pauli matrix is simply a math entity. In terms of measurement, given that one has matrices, there are also commutation relations which are important. One may consider another math operator linked with rotational invariance, namely d/dphi for a rotation about a z-axis. In classical physics, one does not have such a derivative appearing as one deals with functions. If d/dphi is an operator generating rotations, then d/dx should generate translations, but this is simply math. The “catch” seems to be that if one introduces a probability f(x,p) (which is unusual because P(x)dx = dx/L, where L is an arbitrary length), then f(p1,x)f(p2,x)=f(p3,x)f(p4,x) if p1+p2=p3+p4 (we use 1-dimension for simplicity). One may, however, shift this equation to x+dx which implies that df(p1)/dx f(p2) + f(p1)df(p2)/dx = df(p3)/dx f(p4) +f(p3) df(p4)/dx. A solution which keeps p present is: exp(i C p x). We use i because we want to have equal modulus at all x points. If C=1, the one has exp(i px) with -id/dx as the operator which retrieves p. One may then apply this idea to L = r x p and find that this new operator commutes like the Pauli matrices and gives rise to d/dphi, the rotational operator postulated above. Thus, it seems that one could start with the notion of spin and obtain the notion of exp(-iEt+ipx), we argue and spin being linked to angular momentum, we argue.
In Part I of this note, we argued that the exp(ipx) portion of a free quantum particle’s wavefunction follows from d/dx A(x,t) = p extended to an eigenvalue equation: -id/dx exp(iA) = p exp(iAc) “p” appears as a scalar because it is the momentum magnitude, but one may consider flow in 3-space i.e. px,py,pz. In such a case, one might think of using basis vectors ei such that ei dot ei =1 (i=1,2,3). Given that one has an operator equation, one uses basis operators or matrices instead leading to the Pauli matrices or 4x4 matrices as found by Dirac if one imposes the restriction that upon “squaring” the operators disappear because one only has pp. As a result, we argued that intrinsic spin is linked to 3-space recognition for a free particle. The resulting spin, however, is spin ½ so what about other spins say 0 or 1? This seems to imply that free recognition of 3-space does not apply to other spin cases as it does to spin ½. The Stern-Gerlach experiment already shows that spin ½ is interesting in that one may have spin up or down project along any 3-space axis. In Part I, we showed that a wave-equation developed for the photon from d/dt (partial) b (ElEl + BB) + grad dot ElxB = 0 contains an eijk (-i d/dxi) term with eijk being the Levi-Civita symbol as well as the spin matrix. This is not the Pauli matrix (or matrix containing it) and suggests a different geometry or basis scheme. Thus, there is physical angular momentum (spin) in a plane perpendicular to the momentum and the (1,0) spin component is “hidden”, but associated with angular momentum in another spatial direction. As a result, 3-space is not treated in an independent way as in the spin ½ case. In this note, we investigate this idea further by considering the deuteron, He4 and He3 and try to argue that intrinsic spin still seems to have a connection to 3-space recognition, but that the fully independent 3-space recognition corresponds to spin ½. We only consider spin 0, ½ and 1 in this note.
In Part 0 (Parts I and II deal with spin) of this note, we argued that one may begin with two flow equations: d/dx (partial) A(x,t) = p ((1a)) and d/dt (partial) A(x,t) = -E ((1b)) and find that A(x,t) is the classical action for a free particle i.e A(x,t)= L(v) t where L is the classical Lagranian. In particular, after taking the partial derivatives in ((1a)) and ((1b)) one may set x/t=v. Alternatively, one may link ((1a)) and ((1b)) to eigenvalue equations i.e. id/dt exp(i A) = E exp(iA) ((2a)) and -id/dx exp(iA) = p exp(iA) ((2b)). Thus operators appear, in particular one has the momentum operator -id/dx associated with px etc.. These two equations may be incorporated in EE=pp+mm or E= pp/2m (although E and p have different meanings) to obtain the Klein-Gordon and Schrodinger equations. At this point there is no spin. We argued in Part I that spin may be brought in by considering p in ((1a)) as flowing in 3 directions px, py and pz. In such a case one would normally have pi ei where ei is a basis vector, but with the operator equations being used ((2a)) ((2b)), ei is replaced by a basis operator ai (a matrix) such that when pi ai is squared, ai disappears and one obtains pipi = -d/dx d/dx - d/dy d/dy-d/dz d/dz As P. Dirac has shown this leads to 4x4 matrices containing the 2x2 Pauli matrices. As a result, the independent px,py,pz flow is only linked to spin 1/2 . In part II we discussed other spins. Other spin matrices follow different constraints. In this note, we wish to link the idea of basis operators ai with the idea of angular momentum i.e. rxp. In (1), the spin of a fermion is calculated using current from the Dirac equation J (vector) together with r x J i.e. the classical form of angular momentum. In this note, we wish to show the link between this approach and the basis operator one, which does not on the surface, bring in any idea of angular momentum, are the same.
Date on paper should be 2022 not 2021. In a previous note, we argued that for a free particle there exists a function A(x,t) such that d/dx (partial) A(x,t) = p ((1a)) and d/dt (partial) A(x,t) = -E ((1b)). In particular, we used p=E(v) v for the relativistic case and p=mov. We showed that A(x,t) is the classical action for a free particle Integral dt L = T L with v=x/t before the derivatives are taken. After, x/t is set to v. As a result, there is a nonrelativistic A(x,t)= T .5mvv and a relativistic one A= -mo sqrt(1-vv) (c=1). We argued that ((1a)) and ((1b)) are in the form of flow equations. Energy E flows through space out of a dx region (i.e there is an energy flux Ev) so if x is held constant, the change in time is -E. We also noted that one may obtain both the Schrodinger and Klein-Gordon equations from ((1a)) and ((1b)) (see reference in (1)). As a result, A(x,t) is linked to both the nonrelativistic and relativistic free particle actions, yields free particle quantum mechanical equations and may also be used to obtain the form of special relativity (as discussed in the reference in (1)). In this note, we wish to consider spin which is not present in ((1a)) and ((1b)). For both relativistic and nonrelativistic classical dynamics which are linked to A(x,t) one deals with numbers (scalars) and so no notion of spin arises. In obtaining both the relativistic Klein-Gordon equation and nonrelativistic Schrodinger equation from ((1a)) and ((1b)), one creates an eigenvalue equation i.e. id/dt exp(i Action) = E exp(iAction) ((2a)) and -id/dx exp(i Action) = p exp(i Action) ((2b)) which only hold for the free particle. One may then use E and p in expressions EE = pp + momo (where p=Ev= dAction/dx partial) and E = pp/2m + V(x) where p=mov=dAction/dx (partial). For the free particle, one has ((2a)) and ((2b)) without establishing the Klein-Gordon equation. Given that p=Ev is a vector, one may try to write ((2b)) as a vector, but given that one has operators acting on exp(i Action) one does not use a basis vector ei (ei dot ei=1), but rather a “basis” operator or a spin matrix. Thus, we argue that spin emerges in this way without having to linearize the Klein-Gordon equation. Interestingly, this approach only holds for fermions. For the photon, which is a boson, a Dirac-like equation may be obtained directly from a d/dt (partial) EL EL + BB + b grad dot (El x B) = 0 as shown in (2) (with a,b being constants). This again follows from a flow scheme.
Physics is based on experimental observables. Some of these observables are functions of other observables. For example, mass is an observable associated with inertia or gravity and velocity is classically measured, but the product mov yields nonrelativistic momentum which may be measured independently through an impulse hit with a target. A further issue with measurement is the case of vectors. For a vector pointing in one direction, one may orient the z axis to lie along this direction. If one always does this, momentum appears as a scalar, but one has to continuously rotate the co-ordinate system to account for different orientations. In this note, we are interested in spin which is linked to a kind of rotation and so we focus on the momentum vector, but only consider a function of momentum which yields a linear E, free particle energy. Thus, for the Lorentz invariant: -EE + p dot p = -momo, we consider the nonrelativistic limit: E = p dot p /2mo. In quantum mechanics, a free particle is represented by the complex probability (wavefunction) exp(ip dot r).This function has physical implications as it manifests itself in n-slit interference experiments so it is physical (not a mathematical convenience), but at the same time is responsible for conservation of momentum in x,y,z because Integral dx exp(ip1x) exp(ip2x) = 0 if p1 is not equal to p2. A physical function (property) through orthogonality creates an important relationship, namely conservation of momentum. We suggest that a similar idea may be at work in a relationship linking a linear E (free particle energy) to a function of momentum. We start with a linear E related to a function of p and only focus on the p vector because if it is treated as a number, the co-ordinate axis must rotate to account for its orientation. We only consider a situation in which p(x), p(y) and p(z) behave in an independent manner in the function of p (i.e. in the same manner as p itself) and suggest that instead of using vector analysis, there should exist a physical co-ordinate axis for each projection p(x), p(y) and p(z) if they are independent. We argue that p dot p = p(x)p(x) + p(y)p(y)+p(z)p(z) does not contain cross terms such as p(x)p(y). This is a kind of orthogonality which we associate with a rotational kind of operator linked with the p(i) operator -id/dx(i). In other words, we seek: p vector = generalized number = p(x)sx+p(y)sy+p(z)sz, where s(i) are generalized numbers or matrices, so that when this multiplied by itself, it yields p dot p. It turns out that the s(i) are the 2x2 Pauli matrices. In the case of a photon, there is no nonrelativistic limit and a linear E equation related to p is: E= |p|c. This equation is not independent in p(x), p(y), p(z) i.e. one cannot write p(x)sx+p(y)sy+p(z)sz an obtain E = {p(x)sx+p(y)sy+p(z)sz} c. Thus we look for another equation containing a linear E and a function of p. In this case, we use energy and momentum densities as given by the continuity equation: d/dt partial energy density + d grad dot ElxB =0 where d is a constant and El and B electric and magnetic 3-vectors. As in the above analysis, one wishes to have a momentum operator linked with a kind of rotational operator which allows one to eliminate vector math and use generalized numbers. In such a case, grad in the continuity equation becomes the momentum vector and eijk (Levi-Civita symbol in the cross product) becomes the spin matrix. Thus in both cases, spin governs the form of the function containing momentum which yields a linear E. In other words, it is not momentum alone which yields the relationship between a linear E (free particle energy) and momentum.