ЯДЕРНАЯ ФИЗИКА, 2011, том 74, № 6, с. 955-960
ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
NOTES ON THE NATURE OF SO-CALLED INTRINSIC SYMMETRIES
©2011 O. S. Kosmachev*
Joint Institute for Nuclear Research, Dubna, Russia Received November 22,2010
We show that in some cases intrinsic symmetries are peculiar conversion or reflection of external symmetries. They form the structure of the microparticles and then determine interactions of these particles.
1. INTRODUCTION
The notion "intrinsic property" has the dual sense. First of all it expresses some inalienable property, which is inherent in object in any case. But sometimes it reflects a presence of a hidden property or insufficiently known one. Every manifestation of intrinsic or any other property is result of interaction with external objects. Therefore intrinsic properties are relative notions.
In this article exhaustive analysis of some mathematical objects is represented. All they have physical sense such as the electron spin, lepton wave equations. Exhaustive analysis means so full consideration, that it leaves no possibilities for a continuation of the mathematical analysis and therefore it eliminates the presence of some additional physical characteristics or interpretations. Intrinsic symmetries in below considered examples are the peculiar conversion or reflection of the space—time symmetries and discrete ones.
2. PAULI GROUP AND SPIN OF THE ELECTRON
The problem of the spin nature cannot be considered solved in spite of a long history of spin concept [ 1] and its successful mathematical formalization [2] for the electron. At first [1] electron was called "spinning". Then Pauli [2] named it "magnetic". A question on rotation of point-like particle is beyond reasonable understanding, therefore spin properties were referred to internal or proper characteristics of the particle. Evidently, the physical picture does not become more understandable.
For a start let us appeal to the generally accepted assumption made by Pauli that the spin is the proper angular momentum of the electron, having the quantum nature and not related to motion of the particle as a whole. This definition has not changed at present. In
E-mail: kos@theor.jinr.ru
the case of the electron the proper angular momentum is known to be described by Pauli a matrices:
It is easily checked that a matrices generate the 16th-order group, just Pauli group [3]. We will denote this group as dY. The group has ten conjugate classes. The center of the group contains four elements. The group has eight one-dimensional and two nonequiva-lent two-dimensional irreducible representations. The rank of the group is equal to 3. This means that all three Pauli a matrices are necessary to generate the group.
Let us introduce the following notation [4]: az ay = ai, axaz = a,2, ay ax = a3
and
ax = bi, ay = b2, az = b3.
It can be shown that
bi = aic, b2 = a2c, b3 = a3c, (1)
where c is one of four (I, —I, iI, —iI) elements of the group center and c = axayaz = iI. Here I is the unit 2 x 2 matrix. This means that operators a1, a2, a3 are connected with operators b1, b2, b3 by simple relations for the given irreducible representation
bi = iai, b2 = ia2, b3 = ia3. (2) It can also be noted that
a2aia= a-i = af, (3)
222 aa = a3, ai — a2 — a3.
This means that elements ai, a2 generate the quaternion subgroup [5]. Let us denote it as — Q2[ai,a2j.
Assuming that elements of the group dY are generators of some algebra, we obtain the following commutation relations for the elements of the algebra
(infinitesimal operators of the proper Lorentz group representation)
[ai,a2]=2a3, [02,03] = 2ai, (4)
[03,01] = 202,
[61,62] = -203, [62,63] = -2ai, [63,61] = -202, [ai,6i]=0, [02,62] =0, [03,63] = 0,
[01.62] = 263, [ai, 63] = -262,
[02.63] = 26i, [02,6i] = -263,
[03, 6i] = 262, [03,62] = —26i.
The obtained commutative relations coincide with commutative relations of the infinitesimal matrices of the proper homogeneous Lorentz group [6] to the factor 2 common for all equalities. Due to construction of commutative relation (4), all six operators 0i, 02, 03 and 6i, 62, 63 have a definite physical meaning.
It follows from the first row of commutative relations (4) that elements 0i, 02, 03 and all their products form the subgroup of three-dimensional rotations. As it follows from the derivation of commutative relations [6], 6i = ax, 62 = ay, 63 = az have the sense of infinitesimal operators of Lorentz transformations.
Taking into account the anticommutation of the operators 6i, 62, 63, the second upper row of commutative relations (4) takes the form:
6i 62 = -03, 6263 = -0i, 636i = -02. (5)
All three equalities express in infinitesimal form the rotation by some fixed angel of one inertial system with respect to another at their relativistic motion [7]. Obviously, upon deviation from uniform rectilinear motion, this effect has a more complex nature. Upon transition to regular repeated motion, for example, to orbital motion, the rotation also becomes regular, i.e., is manifested as rotation. That is why only the sum of orbital momentum and the spin can be the integral of motion of the particle moving along the orbit, rather than orbital momentum and the spin separately.
Thus the analysis of ^-matrix group on the base of commutative relations (4) which are the direct corollary of the Lorentz transformations demonstrates that so-called proper momentum of the particle with the spin equal 1/2 is the consequence of a definite character of motion of this particle, which is not free or rectilinear. This conclusion is in agreement with the well-known fact. It is impossible to measure magnetic moment of the electron related with spin momentum, if it moves freely [ 18].
The explicit form of the operators 0i, 02, 03 and 6i, 62, 63 for irreducible representations allows to evaluate two weight numbers (l0, li), which specify uniquely irreducible representations of the Lorentz
group. Calculation of the eigenvalue for the standard CT-matrice yields l0 = 1/2. We see, that first weight number (l0) coincides with spin value.
The value of the first weight number is determined formally by operators 0i, 02, 03, i.e. by the subgroup of three-dimensional rotations. But generation of the spin rotation is impossible without relativity, as it follows from the above mentioned. It is undoubtedly truly to correlate spin with quite a definite quantum number, if the quantum numbers are interpreted as indices of groups [9]. But it is an unprovable assumption to endow the spin notion with a physical value which exists separately from the motion of the electron as a whole.
Thus two sides of the spin conception has been demonstrated. The first is related to the form of equation. It determines the fermion or boson type of particles. As it is shown above, this side stems from the three-dimensional rotation subgroup. This yields the strict fixing of the spin value as the integer or half-integer constant, rather than physical quantity. It is this side of spin notion that is present in the spin definition proposed by Pauli.
The second side is related to the occurrence of the physical quantity of the spin momentum of the electron (and corresponding magnetic momentum) in the interaction resulting in nonuniform motion. If the motion becomes periodical, repeated, we obtain the particle spin as the physical quantity. In this manifestation (according to the second row of commutation relations (4)) spin is no more a strictly fixed constant. The circumstance that the first side related to the form of equation is initial obligatory and independent of the second becomes fundamental. The second side is realized only in the presence of the nonuniform motion and depends on the first one in its manifestations and details. This can influence significantly the analysis of compound systems or particles with internal structure.
From the generally accepted formalism of the spin equal 1/2 without any additional assumption, we obtain one of the irreducible representations of the Lorentz group and, as a corollary, a physical interpretation of Pauli a matrices. Strictly speaking, it is applicable for description of electrons or objects whose structures are not taken into account.
Consequent and more detailed examination of Pauli group structure (dY) show that it has duality. It means that apart from subgroup Q2[a1,a2] it contains one more subgroup of order eight — q2[a1,a'2]. Defining relations between the generators are the same for both groups. Difference is the order of generators. Both generators of Q2[a1,a2] has fourth order. One generator (a1)ofq2[a1 ,o!2] has fourth order
and another is of second one. Commutation relation for q2[a1,a2] (Lie algebra) has the form:
[a1,a/2 ] = 2a3, K,a3] = -2ai, (6) [a;3,ai] = 2a2,
where a1 = azay, a'2 = a2c, a3 = a1a'2, c = axayaz = = il.
Let us call q2[a1,a'2] as a quaternion group of the second kind. It is not difficult to show, that Q2[a1 ,a2] is related to SU(2) with detU = 1, whereas q2[a1 ,a'2] is related to SU(2) with detU = -1.
If we extend q2[a1,a'2] by the same element as previously c = axayaz = il, we obtain the following commutation relations:
[a1,a2 ] = 2a3, K,a3] = -2ab (7) [a3,a1] = 2a2,
[61,62 ] = -2a3, [62,63 ]=2a1, [63,61] = -2a2, [a1,6/1]=0, [a'2,62] =0, [a3,63] = 0,
[a1,62] = 263, [a1,63] = -262,
[a2,63] = -261, [a2,61 ] = -263, [a3,61] = 262, [a'3,6'2 ] = 261, where 61 = a1 c, 6'2 = a^c, 6'3 = a'3c.
These relations are different from those written above (4). We will connect them with group fY, taken into account that fY and dY are isomorphic.
The representation (7) is called P-conjugate with respect to dY since distinctions appear at the level of the three-dimensional rotation subgroup, i.e. at the first row. The transition from (4) to (7) is equivalent to the following change a2 — ia'2. Due to the definition of a3 we obtain a3 — ia'3. All further deviations from commutation relations (4) in more lower row
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