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.
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.
Spin ½, which follows from Dirac’s linearization of the Klein-Gordon equation, is associated with a fermion which in turn is linked to an antisymmetric spin wavefunction under the permutation of particles if other quantum numbers are the same, e.g. energy etc. We argue that the Klein-Gordon equation with a rest mass mo implies a rest frame in which one may observe rotations If one wishes to linearize E, one may write: E = +/- sqrt( p dot p +momo) (c=1). Here p dot p is rotationally invariant, but E and px, py and pz are treated in a different manner even though E and (px,py,pz) appear equally as vector entries of a 4-vector. As a result, one may attempt to linearize the Klein-Gordon equation -EE + pp = -momo to have E and px,py,pz appear on the same footing. This, however, cannot be done using numbers, but requires 4x4 matrices containing 2x2 Pauli matrices as shown by Dirac. We consider the simpler case here of the Pauli 2x2 matrices such that: A = sx p(x) + sy p(y) + sz p(z) and AA = -EE+p dot p. The Pauli matrices are 2x2 and ensure that there is orthogonality among p(x), p(y), p(z) so no cross terms appear in AA. The 2x2 Pauli matrices, however, are linked with more than orthogonality of p(x), p(y), p(z). Given that sxsx=sysy=szsz=1, they carry the same feature of the sqrt function, namely that -1 or 1 squared yields 1. In this case, if one considers the two eigenvectors of sz, namely (1,0) and (0,1), the second has an eigenvalue of -1. One must operate twice with sz i.e. sz sz in order to restore 1. Thus sz is a kind of square root rotation. So far, there has been no mention of antisymmetry which requires at least two particles. If one considers W(spin)= |+>|-> + a |->|+>, where |+>=(1,0) and |->=(0,1), (where a could be +1 or -1), one may ask: Is there a condition on an operator which permutes the first particle and second? Here |n>|m> applies to the first particle in state n and the second in m. We consider the operator sz(1) sz(2), where 1 and 2 denote particles 1 and 2, acting on W(spin). For a single particle, sz acting on |-> returns a -1 eigenvalue, and so for a mixed |+.>|->, an eigenvalue of -1 is also returned. We argue that this represents a physical situation and so the permutation operator should also return the same physical state because Permutation operator * Permutation operator = 1. Thus, we suggest that a permutation should have the same effect as sz(1)sz(2). This suggests that a=-1 i.e. one has |+>|->-|->|+> in order that the permutation operator have the same effect as sz(1)sz(2).
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.
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 Part I, we argued that the square root of a positive real number bb has a degeneracy, i.e. -b and +b are solutions. Usually one adds +/- by hand, but we argued that there may exist a mathematical formalism which automatically carries this degeneracy. We suggested this formalism involves the three 2x2 Pauli matrices, each of which has eigenvalues -1 and +1 associated with this degeneracy for 3-vector components. Here we suggest that a positive number bb upon which the square root acts, may be thought of as a vector dot product. In other words, b is the vector magnitude, but in a rotated frame one would have (bx,by,bz) and each of these carry a degeneracy of +/-. A dot product is formally defined as: Sum i,j bi bj delta(i,j), but this equation forces one to add +/- by hand when taking the square root. It simply ensures that there is no mixing of i with j (for i not equal to j). The relationship for Pauli matrices: { s(i), s(j)} = 2 delta(i,j) I ((A)) (I = identity matrix and {} means anticommutation) may be used to show that the dot product has removed information related to the +1/-1 degeneracy..In other words, information inherent to a Pauli matrix, namely its eigenvalues of +1 and -1 disappear in ((A)) and so I the 2x2 identity matrix has 2 eigenvectors with the same +1 eigenvalue. This is related to the product of two identical Pauli matrices so +1 * +1 = -1*-1=1. In other words, delta(i,j) used in the dot product explicitly loses degeneracy (+/-) information.
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 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.