Spin as a statistical degree of freedom

Spin is generally believed to be a phenomenon of quantum-theoretic origin. For a long period of time, following Dirac’s derivation of his relativistic equation, it was also believed to be essentially of relativistic origin. This has changed since the work of [60], [41], [5], [19], [54] and others, who showed that spin may be derived entirely in the framework of non-relativistic QT without using any relativistic concepts. Thus, a new derivation of non-relativistic QT like the present one should also include a derivation of the phenomenon of spin. This will be done in this and the next two sections.

A simple idea to extend the present theory is to assume that sometimes - under certain external conditions to be identified later - a situation occurs where the behavior of our statistical ensemble of particles cannot longer be described by $ρ, S$ alone but requires, e.g., the double number of field variables; let us denote these by $ρ1, S1, ρ2, S2$ (we restrict ourselves here to spin one-half). The relations defining this generalized theory should be formulated in such a way that the previous relations are obtained in the appropriate limits. One could say that we undertake an attempt to introduce a new (discrete) degree of freedom for the ensemble. If we are able to derive a non-trivial set of differential equations - with coupling between $ρ1, S1$ and $ρ2, S2$ - then such a degree of freedom could exist in nature.

Using these guidelines, the basic equations of the generalized theory can be easily formulated. The probability density and probability current take the form $ρ = ρ1 + ρ2$ and $⃗ ⃗ ⃗ j = j1 + j2$ , with $⃗ ji$ ( $i = 1, 2$ ) defined in terms of $ρi, Si$ exactly as before (see section 2). Then, the continuity equation is given by

( ) ˜ ˜ ∂-(-ρ1--+---ρ2--) --∂--- ρ1--∂-S1--- ρ2--∂--S2-- + + = 0, ∂ t ∂ x m ∂ x m ∂ x l l l

(56)

where we took the possibility of multi-valuedness of the “phases“ already into account, as indicated by the notation $˜ Si$ . The statistical conditions are given by the two relations

---- d ----- pk ----xk = ---- (57) dt m -------------------- -d------ (T ) pk = F (x, p, t ), (58) dt k

which is required as a consequence of our larger number of dynamic variables. Eq. (59) is best explained later; it is written down here for completeness. The forces $(T ) F k (x, p, t)$ and $(R ) F k (x, p, t)$ on the r.h.s. of (58) and (59) are again subject to the “statistical constraint“, which has been defined in section 3. The expectation values are defined as in (9)-(11).

Performing mathematical manipulations similar to the ones reported in section 2, the l.h.s. of Eq. (58) takes the form

∫ ˜ ˜ -d------ 3 ∂--ρ1- ∂-S1--- ∂-ρ2--∂--S2-- pk = d x [ + dt ∂ t ∂ xk ∂ t ∂ xk ∂ ρ ∂ ˜S ∂ ρ ∂ S˜ - ----1------1- - ----2------2--+ ρ S˜ (1 ) + ρ ˜S (2 ) ], 1 [j,k ] 2 [j,k ] ∂ xk ∂ t ∂ xk ∂ t

(60)

where the quantities $˜ (i ) S , i = 1, 2 [j,k ]$ are defined as above [see Eq. (6)] but with $˜ S$ replaced by $˜ Si$ .

Let us write now $˜ S$ in analogy to section 2 in the form $˜ ˜ Si = Si + Ni$ , as a sum of a single-valued part $Si$ and a multi-valued part $˜ Ni$ . If $˜ N1$ and $˜ N2$ are to represent an external influence, they must be identical and a single multi-valued part $˜ ˜ ˜ N = N1 = N2$ may be used instead. The derivatives of $˜ N$ with respect to $t$ and $xk$ must be single-valued and we may write

∂ S˜ ∂ S ∂ ˜S ∂ S e ----i- = -----i + eΦ, -----i- = -----i-- -- A , k ∂ t ∂ t ∂ xk ∂ xk c

(61)

using the same familiar electrodynamic notation as in section 2. In this way we arrive at eight single-valued functions to describe the external conditions and the dynamical state of our system, namely $Φ, A k$ and $ρi, Si$ .

In a next step we replace $ρi, Si$ by new dynamic variables $ρ, S, ϑ, φ$ defined by

2 ϑ-- ℏ-- ρ1 = ρ cos , S1 = S + φ, 2 2 2 ϑ-- ℏ-- ρ2 = ρ sin , S2 = S - φ. 2 2

(62)

A transformation similar to Eq. (62) has been introduced by [65] in his reformulation of Pauli’s equation. Obviously, the variables $S, ρ$ describe ’center of mass’ properties (which are common to both states $1$ and $2$ ) while $ϑ, φ$ describe relative (internal) properties of the system.

The dynamical variables $S, ρ$ and $ϑ, φ$ are not decoupled from each other. It turns out (see below) that the influence of $ϑ, φ$ on $S, ρ$ can be described in a (formally) similar way as the influence of an external electromagnetic field if a ’vector potential’ $⃗A (s )$ and a ’scalar potential’ $ϕ (s )$ , defined by

ℏc ∂ φ ℏ ∂ φ (s ) ---- ------ (s ) ---- ----- A l = - cos ϑ , ϕ = cos ϑ , 2e ∂ xl 2e ∂ t

(63)

are introduced. Denoting these fields as ’potentials’, we should bear in mind that they are not externally controlled but defined in terms of the internal dynamical variables. Using the abbreviations

(s ) (s ) Aˆ = A + A , ϕˆ = ϕ + ϕ , l l l

(64)

the second statistical condition (58) can be written in the following compact form

∫ ∑ 3 ∂ ρ ∂ S 1 ∂ S e 2 - d x -----[( ----- + e ˆϕ ) + ------ (-------- --Aˆj ) ] ∂ x ∂ t 2m ∂ x c l j j ∫ ˆ ˆ ˆ ˆ 3 e- ∂--Al-- ∂-Aj--- e- ∂-Al--- -∂-ϕ-- + d x ρ [ - vj ( - ) - - e ] c ∂ xj ∂ x c ∂ t ∂ x -------------------- ∫ l l = F (T )(x, p, t) = d3x ρF (T )(x, p, t ), l l

(65)

which shows a formal similarity to the spinless case [see (14) and (24)]. The components of the velocity field in (65) are given by

-1-- -∂-S--- e- vj = ( - Aˆj ). m ∂ xj c

(66)

If now fields $El, Bl$ and $(s ) (s ) E l , B l$ are introduced by relations analogous to (23), the second line of (65) may be written in the form

∫ e e d3x ρ [(e E⃗ + --⃗v × B⃗ ) + (e E ⃗(s) + --⃗v × B ⃗(s )) ], l l c c

(67)

which shows that both types of fields, the external fields as well as the internal fields due to $ϑ, φ$ , enter the theory in the same way, namely in the form of a Lorentz force.

The first, externally controlled Lorentz force in (67) may be eliminated in exactly the same manner as in section 3 by writing

-------------------- ∫ ∫ (T ) 3 e 3 (I ) F (x, p, t) = d x ρ (e E⃗ + --⃗v × B⃗ ) + d x ρF (x, p, t ). l c l l

(68)

This means that one of the forces acting on the system as a whole is again given by a Lorentz force; there may be other nontrivial forces $F (I)$ which are still to be determined. The second ’internal’ Lorentz force in (67) can, of course, not be eliminated in this way. In order to proceed, the third statistical condition (59) must be implemented. To do that it is useful to rewrite Eq. (65) in the form

∫ ∂ ρ ∂ S 1 ∑ ∂ S e 3 ------ ----- ˆ ------ ------- -- ˆ 2 - d x [( + e ϕ ) + ( - Aj ) ] ∂ xl ∂ t 2m ∂ xj c j ∫ 3 ℏ-- -∂-ϑ-- ∂-φ-- -∂-φ--- ∂-φ--- ∂-ϑ-- -∂-ϑ--- + d x ρ sin ϑ ( [ + vj ] - [ + vj ]) 2 ∂ xl ∂ t ∂ xj ∂ xl ∂ t ∂ xj ------------------- ∫ (I ) (I ) = F (x, p, t) = d3x ρF (x, p, t ), l l

(69)

using (67), (68) and the definition (63) of the fields $(s) A l$ and $(s ) ϕ$ .

We interpret the fields $φ$ and $ϑ$ as angles (with $φ$ measured from the $y -$ axis of our coordinate system) determining the direction of a vector

ℏ-- ⃗s = ( sin ϑ sin φ ⃗ex + sin ϑ cos φ ⃗ey + cos ϑ ⃗ez ), 2

(70)

of constant length $ℏ- 2$ . As a consequence, $⃗s˙$ and $⃗s$ are perpendicular to each other and the classical force $⃗ (R ) F$ in Eq. (59) should be of the form $⃗ D × ⃗s$ , where $⃗ D$ is an unknown field. In contrast to the ’external force’, we are unable to determine the complete form of this ’internal’ force from the statistical constraint [an alternative treatment will be reported in section 8] and set

⃗ (R ) --e---⃗ F = - B × ⃗s, mc

(71)

where $⃗ B$ is the external ’magnetic field’, as defined by Eq. (23), and the factor in front of $⃗ B$ has been chosen to yield the correct $g -$ factor of the electron.

for particle variables $ϑ (t ), φ (t)$ describes the rotational state of a classical magnetic dipole in a magnetic field, see [60]. Recall that we do not require that (72) is fulfilled in the present theory. The present variables are the fields $ϑ (x, t ), φ (x, t)$ which may be thought of as describing a kind of ’rotational state’ of the statistical ensemble as a whole, and have to fulfill the ’averaged version’ (59) of (72).

Performing steps similar to the ones described in I (see also section 2), the third statistical condition (59) implies the following differential relations,

∂ φ e 1 ( ) φ˙ + vj ------- = ------------- Bz sin ϑ - By cos ϑ cos φ - Bz cos ϑ sin φ ∂ x mc sin ϑ j cos φ sin φ + --------G - --------G , (73) 1 2 sin ϑ sin ϑ ∂ ϑ e ( ) G ϑ˙ + v ------- = ------ B cos φ - B sin φ - ----3--, (74) j x y ∂ xj mc sin ϑ

and are otherwise arbitrary. The ’total derivatives’ of $φ$ and $ϑ$ in (69) may now be eliminated with the help of (73),(74) and the second line of Eq. (69) takes the form

∫ ∫ 3 ∂ ρ e 3 e ∂ d x ------ -----sj Bj + d x ρ -----sj ------Bj ∂ x mc mc ∂ x ∫ l l ℏ ∂ ϑ ∂ ϑ ∂ φ + d3x ρ ---( cos φ ------G - sin φ ------G + -----G ). 1 2 3 2 ∂ xl ∂ xl ∂ xl

(76)

The second term in (76) presents an external macroscopic force. It may be eliminated from (69) by writing

------------------- ∫ -------------------- (I ) 3 --∂--- (V ) F l (x, p, t) = d x ρ ( - μj Bj ) + F l (x, p, t), ∂ xl

(77)

where the magnetic moment of the electron $μi = - (e ∕mc )si$ has been introduced. The first term on the r.h.s. of (77) is the expectation value of the well-known electrodynamical force exerted by an inhomogeneous magnetic field on the translational motion of a magnetic dipole; this classical force plays an important role in the standard interpretation of the quantum-mechanical Stern-Gerlach effect. It is satisfying that both translational forces, the Lorentz force as well as this dipole force, can be derived in the present approach. The remaining unknown force $⃗ (V ) F$ in (77) leads (in the same way as in section 3) to a mechanical potential $V$ , which will be omitted for brevity.

The integrand of the first term in (76) is linear in the derivative of $ρ$ with respect to $xl$ . It may consequently be added to the first line of (69) which has the same structure. Therefore, it represents (see below) a contribution to the generalized Hamilton-Jacobi differential equation. The third term in (76) has the mathematical structure of a force term, but does not contain any externally controlled fields. Thus, it must also represent a contribution to the generalized Hamilton-Jacobi equation. This implies that this third term can be written as

∫ ∫ 3 ℏ ∂ ϑ ∂ ϑ ∂ φ 3 ∂ ρ ′ d x ρ ---( cos φ ------G1 - sin φ ------G2+ ------G3 ) = d x -----L , 2 ∂ x ∂ x ∂ x ∂ x 0 l l l l

(78)

Collecting terms and restricting ourselves, as in section 5, to an isotropic law, the statistical condition (69) takes the form of a generalized Hamilton-Jacobi equation:

∂ S 1 ∑ ∂ S e ¯L := ( ----- + eϕˆ ) + ------ ( -------- --Aˆ )2 + μ B = L . j i i 0 ∂ t 2m ∂ xj c j

(79)

The unknown function $L 0$ must contain $L ′ 0$ but may also contain other terms, let us write $′ L0 = L 0 + ΔL0$ .

6 Spin as a statistical degree of freedom