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.
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.
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 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.
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 1, we tried to understand why spin arises. The basic conclusion was that it is linked with an intrinsic force which in turn is associated with E,p (energy momentum) and geometry (rotational symmetry and other symmetries). In practical terms, this means finding an equation linear in intrinsic force (as force appears linearly in nature), but still tied to E,p and geometry. Interestingly, this leads to two very different results for particles with rest mass (spin ½) and photons, although both satisfy the same special relativistic equations (albeit the photon with rest mass mo=0) and both have exp(-iEt+ipx) free wavefunctions. In the photon case, the Electric field El is the intrinsic force (together with the magnetic B) which gives rise to both energy and momentum density. Specifically, energy density= .5eo El dot El + .5/uo B dot B and momentum density = 1/uo El x B. In the photon case, however, there is no rest frame and so the classical vector El is directly associated with E (energy) and p (momentum) without one needing to consider special relativity. Instead one directly uses Maxwell’s electromagnetic equations in order to find a linear equation describing the classical vector El. This equation involves d/dt partial and grad partial which are linked to the quantum operators for E and p (id/dt, -igrad) and El itself is proportional on exp(-iEt+ipx) (actually cos(-wt+kx)), also from Maxwell’s equations. Given this scenario, one has a classical vector El which is perpendicular to B and both lie in a plane perpendicular to motion. For circular polarization, El moves in a circle and given that it gives rise to energy, this energy circulates like classical angular momentum. For the 3-vector for El, one says that one has spin 1, but for circular polarized light, this is classical angular momentum. As a result, spin in the photon case follows from the behaviour of the intrinsic force (a linear equation in this force as shown in Part 1), but one must also have a clear link between the intrinsic force and the consequential energy and force, because it is the energy which actually has to rotate. It is not enough to have El rotate. In the case of a particle with rest mass, we argued in Part 1 that mo (Newton’s inertia) is the intrinsic force which together with velocity gives rise to the consequential energy E and momentum p. The situation is that one actually has two physical frames, the rest frame with the intrinsic force mo and the moving frame in which one has E,p. In order to link the intrinsic force mo to E, p, one must use a Lorentz invariant. We suggested in Part 1 that -Et+px = -mocc t1 is associated with -E dt + p dx = -mocc dt1 + (p=0) * (dx1->inifite) =0. Then dx=hbar/p, dt=hbar/E and E/dx and p/dt are average forces as is mocc/dt1. Tthe idea of force is very much the cornerstone of exp(-iEt+ipx) we argue. In order to see spin arise, one needs an equation linear in mo (the intrinsic force), but linked to E,p and showing geometrical invariances, according to the recipe of Part 1. -EE + cc p dot p = -momocc ((1)) links E,p, mo and shows invariances, but is not linear in mo. One must then linearize it. This means that one compares to Lorentz frames, the rest frame containing mo and the moving containing E,p. Both mo and p are linked to forces, but mo is the intrinsic force and p, the consequential. As a result, they must somehow be separated as they are in ((1)), in particular they are perpendicular if one considers: EE = pz pz + momo (c=1, p->px). In order to linearize without using unit vectors, one cannot use numbers, but must use matrices (as Dirac showed). The operator linked with pz must anticommute with that linked with mo. The operators associated with the consequential and intrinsic forces differ and in fact anticommute in the rest mass case. One may show that one requires at least a 2x2 matrix (the full Dirac treatment uses a 4-vector made of 2 spinors), which means that there is a two vector state. This is presumably associated with motion, if exp(ipz z) is linked to motion as well. In order to analyze this 2 vector state (spinor), we first compare the situation with the photon case. There, one had a 3-vector El which means that one knows an exact position in the plane. The second piece of information must be the direction in which this vector is moving in a circle (for circularly polarized light). There are two a priori given pieces of information which together with the link between El and energy density yields classical angular motion. In the rest mass case, one again expects rotational motion in a plane perpendicular to the z axis (because pz describes all motion along z). (1,0) and (0,1), however, describe classical rotational direction, clockwise and counterclockwise, but this spin ½ case cannot be classical rotation because the physical vector in the plane is not given (as it is by El the intrinsic force). Thus, one has only the direction of rotation. We argue that if exp(ipx) means that one has p hits at different x within hbar/p with probability exp(ipx), one may imagine different x,y positions of energy in the plane (with probabilities) as long as the rotation for (1,0) for each is in the clockwise direction (for example). This means that a physical vector may appear at some x,y in the plane and then at an x1,t1 which would be considered a classical time in the past as long as the motion is clockwise. In this way, one may have an angle of 720 degrees (instead of 360 degrees) describe the motion. In terms of interaction with a B field, one has the clear two directions (spin up clockwise and spin down counterclockwise as examples). Thus, spin must exist in the rest mass case, because one must have mo perpendicular to p if special relativity is to hold. If one does not want to have spin, then one must disregard the equivalence of frames moving at constant speeds with respect to each other and that seems unphysical. (Note: There are spin 0 objects (Higgs particle), but we do not consider that here.
After the discovery of id/dt partial and -id/dx partial as energy and momentum operators in quantum mechanics, Dirac linearized the Klein-Gordon equation to obtain 4x4 matrices based on Pauli 2x2 matrices. As a result, a quantum spin degree of freedom emerged from theory, and matched experimental results. Given the Lorentz invariant form: = -EE + p dot p = -momo (c=1), one may mathematically write: E = +/- sqrt( p dot p + momo), which is standard procedure. One may ask: What is wrong with this expression? We argue in this note that it is not incorrect, but rather incomplete mathematics for the reason that one has to insert +/- by hand. We suggest that a complete mathematical description of the square root of a dot product p dot p should automatically give rise to two solutions associated with +/-. Thus, Dirac’s linearization is not simply a procedure for finding spin, but, we suggest, is a more complete mathematical approach to describing the square root of p dot p, which holds even if one does not consider physical situations. Interestingly, nature seems to choose this more complete mathematical description rather than the looser one using the sqrt function and the manually inserted +/- signs.
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.