The Pauli/Dirac Matrices

Transcription

1 The Pauli/Dirac Matrices By Flamenco Chuck Keyser 1/6/17 Flamenco Chuck Latest Revision /1/17 11:1 AM PST This document shows the relationship of the Pauli/Dirac matrices to the relativistic unit circle (I m currently writing it, but this will give you a taste so far). The Relativistic Unit Circle (under revision) There are three Pauli Matrices for the case of two identical particles 1, 1 : 1 1, 1 1 i i, 1 i i 1, 1 i 1, 1 1

2 i j k k j i k ( ) E i j j i k k ( ) () (1 )() Stern=Gerlach Experiment Although the energy represented as null is equal and opposite, it is directed along an axis perpendicular to the direction of motion classically; globally, we call the direction of motion v I, and indicate the equal and opposite attractive forces by ( J) i i 1 i i i E i j i j i i k k 1 i i j j i k k i j i j i i i k k i Here we see there is an extra term () ( ) () that is different from the initial condition of This term represents the energy carried to the system from the B field that was turned on to separate the spins We can add this term to the initial condition by assuming it is not interacting; thus the total system is represented by the interaction charges and a separate interacting B field: i j i j ( )( ) k k j i j i ( )( ) k k i j j i

3 The total of the interactions from and as initial states is then E E (), and is equal to that of the final states: E E 1 ()

4 The Dirac Matrix The Dirac matrix can then be represented from the relativistic unit circle as a ccw Lorentz rotation corresponding to separate rotations of starting with : 1 cos 1 sin 1 i cos 1 i sin where the trace of the upper left quadrant together with the trace of the lower right quadrant yields the Pauli matrix :, 1 1 Tr( A), Tr( B) A cos, sin B cos sin A B This matrix can be reconfigured to represent polarity reversal: 1 cos 1 cos i sin i sin Spin and Angular Momentum quantization is represented by Lorentz rotation to the diagonals, where 1 introduces a factor of, so that an initial Lorentz rotation of corresponds to:

5 1 1 cos 1 1 sin 1 1 cos 1 1 sin

6 CCW Rotation

7 Positive Definite 1 cos 1 sin I 1 i cos 1 i sin Note that the trace of both matrices in the upper left and lower right quadrants is now: Tr( A) Tr( B) cos sin Compare this result with the equation: cos sin cos sin cos sin Note that there is now an interaction term for the positive definite representation of, and. that now must be accounted for in the expansion This is done by the reamaining Dirac gamma matrices, i 1 i i i i i i i i i i i

8 i i 1 i i 1 T i i i i i 1 i i 1 i 1 i 1 i i 1 i i 1 i i i 1

9 1 1 i 1 i 1 i 1 i 1 i 1 1 i i

10 The interaction product The Interaction Product is the area ( energy ) of the interaction triangle A in each quadrant at the null vector at the vector connection (or at the circumference only if equal and opposite as the tangent, since independent on the plane, For 1, there is a phase difference of in each quadrant, and of for each axis considered separately. For each axis separately, with equal and opposite interaction at the origin:,,, phase difference 1 1 For 1, the sum is equal to (1 ) Energy as sum of quadrants For the energy in each quadrant, Lorentz rotating ccw in the relativistic unit circle with phase difference :, 1 st Quadrant i j 1 k j i 1 k i j j i 1 Interaction Product: 1.

11 (, ) nd quadrant i j 1 k j i 1 k i j j i 1 Interaction Product: 1, rd quadrant i j 1 k j i 1 k i j j i 1 Interaction Product 1 1, th quadrant i j 1 k j i 1 k i j j i 1 Interaction Product: E

12 The interaction energy can also be calculated for in each quadrant as the matrix (e.g. in the first quadrant) as 1, consistent with the argument by geometry above. This interaction relation is actually spin polarization energy of photon on photon interaction (with representing a change (or perturbation) to the initial rest energy of the photon. However, if is characterized as the spin of a photonequivalent electron (i.e., a massive heavier photon), then the spin is independent of the electron mass as defined by the relativistic unit circle; if there is no spin interaction with an external field, then the initial state of the electron is the same as the final state ct ct ', 1, and its rest energy or rest mass is invariant. The spin can be quantized; for the case 1 h describing two photon - equivalent electrons (or photons) interaction, the interaction energy is then defined 1 as Es s h for each quadrant of the relativistic circle, so that 1 h s h. For n such interactions, then s n nh ns Consider the case where 1 (i.e., the initial state is invariant). Then ' 1 Since 1 ' in the case of the relativistic unit circle, this becomes, where can be thought of as the photo-equivalent electron. Then for n such electrons, the equation becomes ' n n ) ( n n However, since be written as ' n n1 n( n 1 ), (where 1, the equation can 1 usually appears as 1, so that ' n n), bearing in mind that is always positive if the interaction is between bases in the first quadrant. Note that if, ' n lim ' limn n lim n( n1) n ; i.e., interaction). In particular, n n n (i.e., there is no n is analogous to a continuum in the state model of atomic physics (note that suggests a

13 three dimensional volume, but the precise discussion is outside the scope of this document).. n n n n From the Binomial Expansion, ( ) rem (,, n) For the case n, rem(,,) which is always positive in the first quadrant (and the third), so that it only vanishes if, 1 (there is only one non-interacting photon or photo-equivalent electron).