Spin - Fisher information

7 Spin - Fisher information

Let us summarize at this point what has been achieved so far. We have four coupled differential equations for our dynamic field variables $ρ, S, ϑ, φ$ . The first of these is the continuity equation (64), which is given, in terms of the present variables, by
$[ ] ∂ ρ ∂ ρ ( ∂ S e ) ----- ------ ---- ------- -- ˆ + - Aj = 0. ∂ t ∂ xl m ∂ xj c$ (88)

The three other differential equations, the evolution equations (81), (82) and the generalized Hamilton-Jacobi equation (87), do not yet possess a definite mathematical form. They contain four unknown functions $Gi, L0$ which are constrained, but not determined, by (83), (86).

The simplest choice, from a formal point of view, is $G = L = 0 i 0$ . In this limit the present theory agrees with Schiller’s field-theoretic (Hamilton-Jacobi) version [31] of the equations of motion of a classical dipole. This is a classical (statistical) theory despite the fact that it contains [see (71)] a number $ℏ$ . But this classical theory is not realized in nature; at least not in the microscopic domain. The reason is that the simplest choice from a formal point of view is not the simplest choice from a physical point of view. The postulate of maximal simplicity (Ockham’s razor) implies equal probabilities and the principle of maximal entropy in classical statistical physics. A similar principle which is able to ’explain’ the nonexistence of classical physics (in the microscopic domain) is the principle of minimal Fisher information [11]. The relation between the two (classical and quantum-mechanical) principles has been discussed in detail in I.

The mathematical formulation of the principle of minimal Fisher information for the present problem requires a generalization, as compared to I, because we have now several fields with coupled time-evolution equations. As a consequence, the spatial integral (spatial average) over $˜ ρ (L - L0 )$ in the variational problem (52) should be replaced by a space-time integral, and the variation should be performed with respect to all four variables. The problem can be written in the form
∫ ∫
3 ( ˜ )
δ dt d x ρ L - L0 = 0

Ea = 0, a = S, ρ, ϑ, φ,
(89)

(90)
where $Ea = 0$ is a shorthand notation for the equations (88), (87), (82) (81). Eqs. (89), (90) require that the four Euler-Lagrange equations of the variational problem (89) agree with the differential equations (90). This imposes conditions for the unknown functions $L0, Gi$ . If the solutions of (89), (90) for $L , G 0 i$ are inserted in the variational problem (89), the four relations (90) become redundant and $ρ ( ˜L - L ) 0$ becomes the Lagrangian density of our problem. Thus, Eqs. (89) and (90) represent a method to construct a Lagrangian.

We assume a functional form $L0 (χ α, ∂k χ α, ∂k ∂l χ α )$ , where $χ α = ρ, ϑ, φ$ . This means $L0$ does not possess an explicit $x, t$ -dependence and does not depend on $S$ (this would lead to a modification of the continuity equation). We further assume that $L0$ does not depend on time-derivatives of $χ α$ (the basic structure of the time-evolution equations should not be affected) and on spatial derivatives higher than second order. These second order derivatives must be taken into account but should not give contributions to the variational equations (a more detailed discussion of the last point has been given in I).

The variation with respect to $S$ reproduces the continuity equation which is unimportant for the determination of $L , G 0 i$ . Performing the variation with respect to $ρ, ϑ, φ$ and taking the corresponding conditions (87), (82) (81) into account leads to the following differential equations for $L0, G1 cos φ - G2 sin φ$ and $G3$ ,

(91)

(92)

(93)
The variable $S$ does not occur in (91)-(93) in agreement with our assumptions about the form of $L 0$ . It is easy to see that a proper solution (with vanishing variational contributions from the second order derivatives) of (91)-(93) is given by
[ ( ) ( ) ]
ℏ2 1 ∂ ∂ √ -- 1 ∂ φ 2 1 ∂ ϑ 2
------ ---------------- -- 2 ----- -- -----
L0 = √ -- ρ - sin ϑ -
2m ρ ∂ ⃗x ∂ ⃗x 4 ∂ ⃗x 4 ∂ ⃗x
[ ( ) ]
ℏ2 1 ∂ φ 2 1 ∂ ∂ ϑ
------ -- ----- -- ----- -----
ℏG1 cos φ - ℏG2 sin φ = sin 2 ϑ - ρ
2m 2 ∂ ⃗x ρ ∂ ⃗x ∂ ⃗x
2
ℏ 1 ∂ ( 2 ∂ φ )
ℏG = - ------- ----- ρ sin ϑ ----- .
3 2m ρ ∂ ⃗x ∂ ⃗x
(94)

(95)

(96)
A new adjustable parameter appears on the r.h.s of (94)- (96) which has been identified with $ℏ2 ∕2m$ , where $ℏ$ is again Planck’s constant. This second $ℏ$ is related to the quantum-mechanical principle of maximal disorder. It is in the present approach not related in any obvious way to the previous “classical“ $ℏ$ which denotes the amplitude of a rotation.

The solutions for $G1, G2$ may be obtained with the help of the second condition ( $⃗ G ⃗s = 0$ ) listed in Eq. (83). The result may be written in the form
$( ) ℏ 1 ∂ 1 ∂ φ ∂ ϑ G1 = -------------ρ -- sin 2ϑ sin φ ----- - cos φ ----- 2m ρ ∂ ⃗x 2 ∂ ⃗x ∂ ⃗x ( ) ℏ 1 ∂ 1 ∂ φ ∂ ϑ ------------- -- ----- ----- G2 = ρ sin 2ϑ cos φ + sin φ . 2m ρ ∂ ⃗x 2 ∂ ⃗x ∂ ⃗x$ (97)

Eqs. (96) and (97) show that the first condition listed in (83) is also satisfied. The last condition is also fulfilled: $L 0$ can be written as $L ′ + ΔL 0 0$ , where
$[ ( ) ( ) ] ℏ2 ∂ φ 2 ∂ ϑ 2 ℏ2 1 ∂ ∂ √ -- ′ ------ 2 ----- ----- --------------------- L 0 = - sin ϑ - , ΔL0 = √ -- ρ, 8m ∂ ⃗x ∂ ⃗x 2m ρ ∂ ⃗x ∂ ⃗x$ (98)

and $′ L 0$ fulfills (86). We see that $′ L 0$ is a quantum-mechanical contribution to the rotational motion while $ΔL 0$ is related to the probability density of the ensemble (as could have been guessed considering the mathematical form of these terms). The last term is the same as in the spinless case [see (59)].

The remaining task is to show that the above solution for $L0$ does indeed lead to a (appropriately generalized) Fisher functional. This can be done in several ways. The simplest is to use the following result due to Reginatto [28]:
( )
∫ 2 3 ∫ 3 (j ) 2
ℏ ∑ ∑ 1 ∂ ρ
d3x ( - ρL ) = ------ d3x ------ --------- ,
0 8m (j ) ∂ x
j=1 k=1 ρ k

ϑ φ ϑ φ ϑ
(1) 2 --- 2 --- (2 ) 2 --- 2 --- (3) 2 ---
ρ := ρ sin cos , ρ := ρ sin sin , ρ := ρ cos .
2 2 2 2 2
(99)

(100)
The functions $ρ (j )$ represent the probability that a particle is at space-time point $x, t$ and $⃗s$ points into direction $j$ . Inserting (94) the validity of (99) may easily be verified. The r.h.s. of Eq. (99) shows that the averaged value of $L0$ represents indeed a Fisher functional, which completes our calculation of the ’quantum terms’ $L , G 0 i$ .

Summarizing, our assumption, that under certain external conditions four state variables instead of two may be required, led to a nontrivial result, namely the four coupled differential equations (88), (87), (82) (81) with $L0, Gi$ given by (94), (97), (96). The external condition which stimulates this splitting is given by a gauge field; the most important case is a magnetic field $⃗ B$ but other possibilities do exist (see below). These four differential equations are equivalent to the much simpler differential equation
$( ℏ ∂ ) 1 ( ℏ ∂ e )2 -------+ e φ ψˆ + ------ ------- - --A⃗ ˆψ + μB ⃗σ B⃗ ψˆ = 0, i ∂ t 2m i ∂ ⃗x c$ (101)

which is linear in the complex-valued two-component state variable $ˆ ψ$ and is referred to as Pauli equation (the components of the vector $⃗σ$ are the three Pauli matrices and $μ = - eℏ ∕2mc B$ ). To see the equivalence one writes [34],[14]
$( φ ) ϑ- i-- √ -- i-S cos 2 e 2 ˆψ = ρ e ℏ ( ) , ϑ - iφ- i sin -- e 2 2$ (102)

and evaluates the real and imaginary parts of the two scalar equations (101). This leads to the four differential equations (88), (87), (82) (81) and completes the present spin theory.

In terms of the real-valued functions $ρ, S, ϑ, φ$ the quantum-mechanical solutions (94),(96),(97) for $L0, Gi$ look complicated in comparison to the classical solutions $L0 = Gi = 0$ . In terms of the variable $ˆ ψ$ the situation changes to the contrary: The quantum-mechanical equation becomes simple (linear) and the classical equation, which has been derived by Schiller [31], becomes complicated (nonlinear). It is satisfying that the simplicity of the underlying physical principle (principle of maximal disorder) leads to a simple mathematical representation of the final basic equation.

Besides the Pauli equation we found, as a second important result of our spin calculation, that the following local force is compatible with the statistical constraint:
$( ) 1 ∂ ⃗F L + ⃗F I = e E⃗ + --⃗v × ⃗B - ⃗μ ⋅ -----⃗B. c ∂ ⃗x$ (103)

Here, the velocity field $˜⃗v (x, t)$ [see (34),(35)] and the magnetic moment field $⃗μ (x, t) = - (e ∕mc )⃗s (x, t)$ [see (78),(85)] have been replaced by corresponding particle quantities $⃗v (t )$ and $⃗μ (t )$ ; the dot denotes the inner product between $⃗μ$ and $⃗B$ . The first of the two forces in (103) is the Lorentz force; it has been derived here (basically in the same manner as in the spinless case) from first principles without any additional assumptions. The same cannot be said about the second force which takes this particular form as a consequence of some additional assumptions concerning the form of the ’internal force’ $F⃗ R$ [see (79)]. In particular, the field appearing in $F⃗ R$ was arbitrary as well as the proportionality constant (g-factor of the electron) in front of it. It is well-known that in a relativistic treatment the spin term appears automatically if the potentials are introduced. Interestingly, this unity is not restricted to the relativistic regime but may as well be created in non-relativistic treatments [4],[12]. A simple indication of this unity is the fact that the spin term contains no new fundamental constants. One might speculate that an analogous formulation for the present theory would automatically eliminate the above mentioned shortcoming. A most natural framework to study this point - which is a subject of future research - is probably a relativistic one.

In the present treatment spin has been introduced as a property of an ensemble and not of individual particles. Of course, the properties of an ensemble cannot be thought of as being completely independent from the properties of the particles it is made from. The question whether or not a property ’spin’ can be ascribed to single particles is a subtle one. Formally, we could assign a probability of being in a state $i$ ( $i = 1, 2$ ) to a particle just as we assign a probability for being at a position $3 ⃗x ∈ R$ . The problem is that, contrary to position, no classical meaning - and no classical measuring device - can be associated with the discrete degree of freedom $i$ . Experimentally, the measurement of the ’spin of a single electron’ is - in contrast to the measurement of its position - a notoriously difficult task. Such experiments, and a number of other interesting questions related to spin, have been discussed by Morrison [26].

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