ABSTRACT
Besides continuous transformations (the rotations in the four-dimensional space), the Lorentz transformations also contain two discrete operations, namely, the space and time
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inversions (P and T ). They are defined as follows:
P
x0
x1
x2
x3
=
1 0 0 0 0 −1 0 0 0 0 −1 0 0 0 0 −1
x0
x1
x2
x3
=
x0
−x1 −x2 −x3
, (3.5)
T
x0
x1
x2
x3
=
−1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
x0
x1
x2
x3
=
−x0 x1
x2
x3
. (3.6)
Hereinafter to write down 4× 4-matrices we follow the agreement: Λ ≡ Λµν ,
where the superscript numerates a column, and the subscript indicates a row. The transformations L↓± involve the time reflections, while L
↑ ± do not. Analogously, the transformations
L↓↑− contain the space inversion, while L ↓↑ + do not. The homogeneous Lorentz group can be
symbolically represented as the sum:
L = L↑+ + PTL ↑ + + PL
↑ + + TL
Since any Lorentz transformation can be presented as the product L↑+, P , T , PT , we can limit ourselves to studying the space-space rotations (the rotations in the planes [x1x2], [x1x3] and [x2x3]) and the purely Lorentz transformations (the transformations connecting two inertial reference frames that move with the relative speed v∗) only. The totality of these transformations constitutes a group of the limited homogeneous Lorentz transformations L↑+ (the proper orthochronous Lorentz group or, simply, the proper Lorentz group). If the speed of relative movement of the inertial reference frames (IRFs) under consid-
eration is directed along the common axis x1, then we have the following relation between their coordinates:
x0
x1
x2
x3
′
=
(1− β2)−1/2 −β(1− β2)−1/2 0 0 −β(1− β2)−1/2 (1− β2)−1/2 0 0
0 0 1 0 0 0 0 1
x0
x1
x2
x3
, (3.8)
where β = v/c. Setting:
(1− β2)−1/2 = coshu, β(1 − β2)−1/2 = sinhu, we can interpret the Lorenz boost along the axis x1 as a rotation in the plane x1x0 by the angle u = arctanh β. Actually,
x0
x1
x2
x3
′
= Λ(10;u)
x0
x1
x2
x3
=
coshu − sinhu 0 0 − sinhu coshu 0 0
0 0 1 0 0 0 0 1
x0
x1
x2
x3
. (3.9)
It is not out of space to emphasize the conventional character of such a interpretation of the boost. Since tanu = iβ, the rotation proceeds by the imaginary angle. This approach,
however, allows one to interpret the limited homogeneous Lorenz transformations as the group, which consist only of the rotations in the four-dimensional Minkowski space. Let us determine the explicit form for generators of the Lorentz transformations. A
generator for the rotation in the plane [x1x0] is defined by the formula:
J10 = d
du Λ(10;u)
∣∣∣∣∣ u=0
and has the form
J10 =
0 −1 0 0 −1 0 0 0 0 0 0 0 0 0 0 0
. (3.10)
Analogously, the generators J20 and J30, corresponding to the boosts along the coordinate axes x2 and x3, are given by the expressions:
J20 =
0 0 −1 0 0 0 0 0 −1 0 0 0 0 0 0 0
, J30 =
0 0 0 −1 0 0 0 0 0 0 0 0 −1 0 0 0
. (3.11)
The generators of the space rotations in the four-dimensional notations look as follows:
J12 =
0 0 0 0 0 0 1 0 0 −1 0 0 0 0 0 0
, J23 =
0 0 0 0 0 0 0 0 0 0 0 1 0 0 −1 0
, (3.12)
J31 =
0 0 0 0 0 0 0 −1 0 0 0 0 0 1 0 0
. (3.13)
Since the rotation in the plane [xµxν ] by the angle ϕ is identical to that in the plane [xνxµ] by the angle −ϕ, then the equality: Jµν = −Jνµ takes place. An arbitrary infinitesimal transformation of the homogeneous Lorentz group can be written down in the following way:
Λ(ω) = I + 1
2 ωµνJ
µν , (3.14)
where ωµν = −ωνµ. By means of direct calculations one can be convinced, that the generators Jµν of the Lorentz group obey the permutation relations:
[Jµν , Jρσ ] = gµρJνσ + gνσJµρ − gµσJνρ − gνρJµσ. (3.15)
Suppose, there exists two IRFs: K and K ′ with the coordinates x and x′, respectively. Let one and the same wave field be described by a K-frame observer with the help of functions
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ψi(x) and a K ′-frame observer with the help of functions ψ′l(x
′). Then values of ψi(x) completely define those of ψ′l(x
′), since x and x′ correspond to one and the same event in the space-time. Thus, ψ′l(x
′) must be expressed in terms of ψi(x). Consequently, under transition from the first observer to the second one, transformation of the fields, described by both, occurs according to the law:
ψn′(x′) = Dnk(Λ)ψk(x). (3.16) It should be emphasized, that the matrix D depends on the Lorentz transformation matrix and the indices n and k may have tensor dimensionality. For the sake of simplicity, the law of the field transformation (3.16) can be written down in a more convenient form:
ψn′(x) = Dnk(Λ)ψk(Λ−1x), (3.17) The totality of matrices (or, what is the same, operators) D(Λ), which specify transformations of field components, constitute the Lorentz group representation. The matrix of any representation of the limited homogeneous Lorenz group D(Λ) can be set down as follows:
D(Λ) = I + 1 2 ωµνMµν , (3.18)
whereMµν compose a representation for the Lee-algebra generators, satisfying permutation relations, which are similar to the permutation relations for the Lorentz group generators (3.15):
[Mµν ,Mρσ] = gµρMνσ + gνσMµρ − gµσMνρ − gνρMµσ. (3.19) Finding representations of the limited Lorentz group is reduced to determining all the irreducible matrices, suiting to the relations (3.19). The group under consideration has finite-dimensional and infinite-dimensional irreducible representations. From the point of view of physical applications we are interested in classification of quantities with the finite number of components transformed according to a finite-dimensional representation of the homogeneous Lorentz group. However, one should not go so far to lose an interest in infinite-dimensional representations in general. Thus, unitary infinite-dimensional irreducible representations of the Poincare group deal with the classification of elementary particles. A thoughtful reader can guess at once, that in the last case we are dealing with the presence in a complete set, classifying the representation basis, of such operators that possess a continuous spectrum, not limited from above (as, for example, the momentum operator). In other words, basis vectors of a representation have the finite number of components, while the number of basis vectors as well as eigenvalues of an operator with a continuous spectrum is infinite. Now we consider finite-dimensional representations of the limited homogeneous Lorentz
group, namely the situation, when transformation matrices contain the finite number of rows (or columns, since matrices are square). Let us define the operators
M = (M32,M13,M21), N = (M01,M02,M03). (3.20)
They obey the following permutation relations:
[Mi,Mj] = εijkMk, [Ni, Nj ] = −εijkMk, [Mi, Nj] = εijkNk. (3.21)
Next, we pass on to new quantities according to the formulae:
X = 1
2 (M+ iN), Y =
2 (M− iN). (3.22)
It is easy to show that they satisfy commutation relations to be already quite clear for physical interpretation:
[Xi, Xj ] = iεijkXk, [Yi, Yj ] = iεijkYk, [Xi, Yj ] = 0. (3.23)
Thus, we have demonstrated that infinitesimal operators of the Lorentz group Mµν could be reduced to two independent vector operatorsX and Y to obey the permutation relations for the moment of momentum projection operators. Therefore, we can obtain all irreducible finite-dimensional representations of the homogeneous Lorentz group when the matrices for the moments are taken as X and Y. Since the angular momentum square commutes with its every projection
[X2, Xi] = 0, [Y 2, Yi] = 0, (3.24)
then the operators X2, Y2 are the Casimir operators. It means, as we already know, that they are multiple to the unit operator:
X2 = j(j + 1)I2j+1, Y 2 = k(k + 1)I2k+1, (3.25)
where j and k are arbitrary positive integer or half-integer numbers, I2j+1 (I2k+1) is the identity matrix, containing 2j + 1 (2k + 1) rows and columns. So, every irreducible representation of the limited homogeneous Lorentz group is characterized by two integer or half-integer numbers j and k. This representation denoted by (j, k), has the dimensionality (2j + 1)(2k + 1), since it is stretched over the totality (2j + 1)(2k + 1) of the basis vectors | j,mj ; k,mk > satisfying the eigenvalue equations:
X2 | j,mj ; k,mk >= j(j + 1) | j,mj ; k,mk >, (3.26) X3 | j,mj ; k,mk >= mj | j,mj ; k,mk >, (3.27)
Y2 | j,mj ; k,mk >= k(k + 1) | j,mj ; k,mk >, (3.28) Y3 | j,mj ; k,mk >= mk | j,mj ; k,mk > . (3.29)
Let us find out for what representations complex-conjugated field functions are transformed. Taking into account that
M∗ =M, N∗ = N,
and making use of (3.22), we conclude that at the complex conjugation the replacement:
X→ Y, Y → X (3.30) takes place. So, if | j,mj ; k,mk > is transformed under representation (j, k), then its complex-conjugated field function is transformed under the mirror representation (k, j). Under the space inversion the speed, entering into a boost, reverses its sign v → −v.
