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CHAPTER 4

Representations of Lorentz and Poincaré groups

4.1Symmetry algebras

Relativistic theories, as we have discussed, should be invariant under Lorentz transformations. In addition, experimentally we know that space-time translations also define a symmetry of physical theories. In this chapter, therefore, we will study the symmetry algebras of the Lorentz and the Poincaré groups as well as their representations which are essential in constructing physical theories. But, let us start with rotations which we have already discussed briefly in the last chapter. In studying the symmetry algebras of continuous symmetry transformations, it is sufficient to study the behavior of infinitesimal transformations since any finite transformation can be built out of infinitesimal transformations. Furthermore, the symmetry algebra associated with a continuous symmetry group is given by the algebra of the generators of infinitesimal transformations. It is worth noting here that, for space-time symmetries, the symmetry algebras can be easily obtained from the coordinate representation of the symmetry generators and that is the approach we will follow in our discussions.

4.1.1 Rotation. Let us consider an arbitrary, infinitesimal rotation in three dimensions of the form (repeated indices are summed)

where α^k represents the infinitesimal constant parameter of rotation around the k-th axis (there are three of them). (Let us recall our notation from (1.34) and (1.35) here for clarity. ϵ_ijk denotes the three dimensional Levi-Civita tensor with ϵ₁₂₃ = 1. ϵⁱ_jk = η^iℓϵ_ℓjk, etc.) If we now identify (in the last chapter we had denoted the infinitesimal transformation matrices by ϵⁱ_j, ϵ^µ_ν, but here we denote them by ωⁱ_j, ω^µ_ν in order to avoid confusion with the Levi-Civita tensors)

then, we note that

and that the infinitesimal rotation around the k-th axis (alternatively in the i-j plane) can also be represented in the form

This is, of course, the form of the rotation that we had discussed in the last chapter.

Let us next define an infinitesimal vector operator (also known as the tangent vector field operator) for rotations (an operator in the coordinate basis) of the form

where we have identified M_ij = x_i∂_j − x_j∂_i. It follows now that

In other words, we see from (4.4) that we can write the infinitesimal rotations in the i-j plane also as

Namely, the vector operator, in (4.5) generates infinitesimal rotations and the operators, M_ij, are known as the generators of infinitesimal rotations.

The Lie algebra of the group of rotations can be obtained from the algebra of the vector operators themselves. Thus, we note that

where we have identified

Namely, two rotations do not commute, rather, they give back a rotation. Such an algebra is called a non-Abelian (non-commutative) algebra. Using the form of in (4.5), namely,

we can obtain the algebra satisfied by the generators of infinitesimal rotations, M_ij, from the algebra of the vector operators in (4.8). Alternatively, we can calculate them directly as

This is the Lie algebra for the group of rotations. If we would like the generators to be Hermitian quantum mechanical operators corresponding to a unitary representation, then we may define the operators, M_ij, with a factor of “i”. But up to a rescaling, (4.11) represents the Lie algebra of the group SO(3) or equivalently SU(2). To obtain the familiar algebra of the angular momentum operators, we note that we can define (recall that in the four vector notation J_i = −(J)_i)

which gives the familiar orbital angular momentum operators. Using this, then, we obtain (p, q, r, s = 1, 2, 3)

where in the last step we have used the Jacobi identity for the structure constants of SO(3) or SU(2) (or the identity satisfied by the Levi-Civita tensors), namely,

where we have used the anti-symmetry of the Jacobi identity in the i, j indices. This, in turn, leads to (see (4.12))

The algebra of the generators in (4.11) or (4.13) is, of course, the Lie algebra of SO(3) or SU(2) (or the familiar algebra of angular momentum operators) up to a rescaling.

4.1.2 Translation. In the same spirit, let us note that a constant infinitesimal space-time translation of the form

can be generated by the infinitesimal vector operator (repeated indices are summed)

so that

and we can write

The Lie algebra associated with translations is then obtained from

In other words, two translations commute and the corresponding relation for the generators is

Namely, translations form an Abelian (commuting) group while the three dimensional rotations form a non-Abelian group.

4.1.3 Lorentz transformation. As we have seen in the last chapter, a proper Lorentz transformation can be thought of as a rotation in the four dimensional Minkowski space-time and has the infinitesimal form

where, as we have seen in (3.20), the infinitesimal, constant parameters of transformation satisfy

As in the case of rotations, let us note that if we define an infinitesimal vector operator (see (4.10))

then, we obtain

Therefore, we can think of as the vector operator generating infinitesimal proper Lorentz transformations and the operators, M_µν = −M_νµ, as the generators of the infinitesimal transformations. We also note that we can identify the infinitesimal generators of spatial rotations with (see (4.12))

and the generators of infinitesimal boosts with

As before, we can determine the group properties of the Lorentz transformations from the algebra of the vector operators generating the transformations. Thus,

where, as in the case of rotations (see (4.8) and (4.9)), we have

This shows that the algebra of the vector operators is closed and that Lorentz transformations define a non-Abelian group.

The algebra of the generators can also be calculated directly and has the form

This, therefore, gives the Lie algebra associated with Lorentz transformations. As we have seen these transformations correspond to rotations, in this case, in four dimensions and, therefore, the Lie algebra of the generators is isomorphic to that of the group SO(4). In fact, we note that the number of generators for SO(4) which is (for SO(n), it is )

coincides exactly with the six generators we have (namely, three rotations and three boosts). However, since the rotations are in Minkowski space-time whose metric is not Euclidean it is more appropriate to identify the Lie algebra as that of the group SO(3, 1). (Namely, Lorentz transformations (boosts) are non-compact unlike rotations in Euclidean space.)

We end this section by pointing out that the algebra in (2.110) coincides with (4.30) (up to a scaling). This implies that, up to a scaling, the matrices σ_µν provide a representation for the generators of the Lorentz group. This is what we had seen explicitly in (3.71) in connection with the discussion of covariance of the Dirac equation.

4.1.4 Poincaré transformation. If, in addition to infinitesimal Lorentz transformations, we also consider infinitesimal translations, the general transformation of the coordinates takes the form

where ϵ^µ, ω^µ_ν denote respectively the parameters of infinitesimal translation and Lorentz transformation. The transformations in (4.32) are known as the (infinitesimal) Poincaré transformations or the inhomogeneous Lorentz transformations. Clearly, in this case, if we define an infinitesimal vector operator as

then, acting on the coordinates, it generates infinitesimal Poincaré transformations. Namely,

The algebra of the vector operators for the Poincaré transformations can also be easily calculated as

where we have identified

We can also calculate the algebra of the generators of Poincaŕe group. We already know the commutation relations [M_µν, M_λρ] as well as [P_µ, P_ν] (see (4.30) and (4.21)). Therefore, the only relation that needs to be calculated is the commutator between the generators of translation and Lorentz transformations. Note that

which simply shows that under a Lorentz transformation, P_µ behaves like a covariant four vector. (This is seen by recalling that ω^µνM_µν generates infinitesimal Lorentz transformations. The commutator of a generator (multiplied by the appropriate transformation parameter) with any operator gives the infinitesimal change in that operator under the transformation generated by that particular generator. For change in the coordinate four vector under an infinitesimal Lorentz transformation, see, for example, (4.22) and (3.59).)

Thus, combining with our earlier results on the algebra of the translation group, (4.21), as well as the homogeneous Lorentz group, (4.30), we conclude that the Lie algebra associated with the Poincaré transformations (inhomogeneous Lorentz group) is given by

We note that the algebra of translations defines an Abelian sub-algebra of the Poincaré algebra (4.38). However, since the generators of translations do not commute with the generators of Lorentz transformations, Poincaré algebra cannot be written as a direct sum of those for translations and Lorentz transformations. Namely,

Rather, it is what is known as a semi-direct sum of the two algebras. (The general convention is to denote groups by capital letters while the algebras are represented by lower case letters.)

4.2Representations of the Lorentz group

Let us next come back to the homogeneous Lorentz group and note that the Lie algebra in this case is given by (4.30)

We recall from (4.12), (4.26) and (4.27) that we can identify the angular momentum and the boost operators as

Written out in terms of these generators, the Lorentz algebra takes the form

where we have used (4.13) in the last relation.

This is a set of coupled commutation relations. Let us define a set of new generators as linear superpositions of J_i and K_i as (this is also known as changing the basis of the algebra)

which also leads to the inverse relations

Parenthetically, let us note from the form of the algebra in (4.42) that we can assign the following hermiticity properties to the generators, namely,

This unconventional hermiticity for J_i arises because, in choosing the coordinate representation for the generators, we have not been particularly careful about choosing Hermitian operators. As a consequence of (4.45), we have

amely, the generators in the new basis are all anti-Hermitian. The opposite hermiticity property of the generators of boosts, K_i, (compared to J_i) is connected with the fact that such transformations are non-compact and, consequently, the finite dimensional representations of boosts are non-unitary (hence the opposite Hermiticity of K_i). However, infinite dimensional representations are unitary, as can be seen from the hermiticity of the generators in the coordinate basis, namely, if we define the generators with a factor of “i”,

In the new basis (4.43), the Lorentz algebra (4.42) takes the form

In other words, in this new basis, the algebra separates into two angular momentum algebras which are decoupled. Mathematically, one says that the Lorentz algebra is isomorphic to the direct sum of two angular momentum algebras,

Incidentally, as we have already seen in the last chapter, the Lorentz group is double valued (doubly connected). Therefore, it is more meaningful to consider the simply connected universal covering group of SO(3, 1) which is to describe the Lorentz transformations, much the same way we consider the universal covering group SU(2) of SO(3) to describe rotations.

The finite dimensional unitary representations of each of the angular momentum algebras are well known. Denoting by j_A and j_B the eigenvalues of the Casimir operators A² and B² respectively for the two algebras, we have

An irreducible nonunitary representation of the homogeneous Lorentz group, therefore, can be specified uniquely once we know the values of j_A and j_B and is labelled as D^(jA,jB) (just as the representation of the rotation group is denoted by D^(j)). Namely, this represents the operator implementing finite transformations on the Hilbert space of states or wave functions as

where Λ represents the finite Lorentz transformation parameter. (Note that we can write D^(jA,jB) = D^(jA)D^(jB), which is obvious in the first line of the following equation (4.52), since the operators A_i commute with B_i.) Explicitly, we can write (this is the generalization of the S(Λ) matrix that we studied in (3.37) in connection with the covariance of the Dirac equation)

where the finite parameters of rotation and boost can be identified with

Such a representation labelled by (j_A, j_B) will have the dimensionality (since it is a product representation)

and its spin content follows from the fact that (see (4.44))

Consequently, from our knowledge of the addition of angular momenta, we conclude that the values of the spin in a given representation characterized by (j_A, j_B) can lie between

The first few low lying representations of the Lorentz group are as follows. For j_A = j_B = 0, we see from (4.54) and (4.56) that

which corresponds to a scalar representation with zero spin (and acts on the wave function of a Klein-Gordon particle). Similarly, for j_A = j_B = 0,

corresponds to a two component spinor representation with spin We note that, for j_A = 0,

which also corresponds to a two component spinor representation with spin These two representations are inequivalent and, in fact, are complex conjugates of each other and can be identified to act on the wave functions of the two kinds of massless Dirac particles (Weyl fermions) we had discussed in the last chapter. For

is known as a four component vector representation and can be identified with a spin content of 0 and 1 for the components. (Note that a four vector such as x^µ has a spin zero component, namely, t and a spin 1 component x (under rotations) and the same is true for any other four vector.) It may be puzzling as to where the four component Dirac spinor fits into this description. It actually corresponds to a reducible representation of the Lorentz group of the form

This discussion can similarly be carried over to higher dimensional representations.

4.2.1 Similarity transformations and representations. Let us now construct explicitly a few of the low order representations for the generators of the Lorentz group. To compare with the results that we had derived earlier, we now consider Hermitian generators by letting M_µν → iM_µν as in (4.47). (Namely, we scale all the generators J_i, K_i, A_i, B_i by a factor of i.)

From (4.50), we note that for the first few low order representations, we have (we note here that the negative sign in the spin representation in (4.62) arises because in (4.48))

Using (4.44), this leads to the first two nontrivial representations for the angular momentum and boost operators of the forms

and

Equations (4.63) and (4.64) give the two inequivalent representations of dimensionality 2 as we have noted earlier. Two representations are said to be equivalent, if there exists a similarity transformation relating the two. For example, if we can find a similarity transformation S leading to

then, we would say that the two representations and are equivalent. In fact, from (4.63) and (4.64) we see that the condition (4.65) would require the existence of an invertible matrix S such that

which is clearly impossible. Therefore, the two representations labelled by and are inequivalent representations. They provide the representations of angular momentum and boost for the left-handed and the right-handed Weyl particles.

From (4.63) and (4.64), we can obtain the representation of the Lorentz generators for the reducible four component Dirac spinors as

However, we note that these do not resemble the generators of the Lorentz algebra defined in (3.71) and (2.99) (or (3.73) and (3.80)). This puzzle can be understood as follows. We note that in the Weyl representation for the gamma matrices defined in (2.120),

As a result, we note that the angular momentum and boost operators in (4.67) are obtained from

and, consequently, give a representation of the Lorentz generators in the Weyl representation. On the other hand, if we would like the generators in the standard Pauli-Dirac representation (which is what we had used in our earlier discussions), we can apply the inverse similarity transformation in (2.122) to obtain

Therefore, we note that the generators in (4.67) and in our earlier discussion in (3.71) and (2.99) (see also (3.73) and (3.80)) are equivalent since they are connected by a similarity transformation that relates the Weyl representation of the Dirac matrices in (2.120) to the standard Pauli-Dirac representation.

There is yet another interesting example which sheds light on similarity transformations between representation. For example, from the infinitesimal change in the coordinates under a Lorentz transformation (see, for example, (3.20)), we can determine a representation for the generators of the Lorentz transformations belonging to the representation for the four vectors. On the other hand, as we discussed earlier, from the Lie algebra point of view the four vector representation corresponds to j_A = j_B = (see (4.60)) and we can construct the representations for J and K in this case as well from a knowledge of the addition of angular momenta. Surprisingly, the two representations for the generators constructed from two different perspectives (for the same four vector representation) appear rather different and, therefore, there must be a similarity transformation relating the two representations. Let us illustrate this for the simpler case of rotations. The case for Lorentz transformations (boosts) follows in a parallel manner.

Let us consider a three dimensional infinitesimal rotation of coordinates around the z-axis as described in (3.5). (Here we will use 3-dimensional Euclidean notation without worrying about raising and lowering of the indices.) Representing the infinitesimal change in the coordinates as

we can immediately read out from (3.5) the matrix structure of the generator J₃ to be

Similarly, considering infinitesimal rotations of the coordinates around the x-axis and the y-axis respectively, we can deduce the matrix form of the corresponding generators to be

It can be directly checked from the matrix structures in (4.72) and (4.73) that they satisfy

and, therefore provide a representation for the generators of rotations. This is, in fact, the representation in the space of three vectors which would correspond to j = 1.

On the other hand, it is well known from the study of the representations of the angular momentum algebra that the generators in the representation j = 1 have the forms¹

which look really different from the generators in (4.72) and (4.73) in spite of the fact that they belong to the same representation for j = 1. (The superscript (LA) denotes the standard representation obtained from the study of the Lie algebra.) This puzzle can be resolved by noting that there is a similarity transformation that connects the two representations and, therefore, they are equivalent.

To construct the similarity transformation (which actually is a unitary transformation), let us note that the generators obtained from the Lie algebra are constructed by choosing the generator J₃^(LA) to be diagonal. Let us note from (4.72) that the three normalized eigenstates of J₃ have the forms

Let us construct a unitary matrix from the three eigenstates in (4.76) which will diagonalize the matrix J₃,

If we now define a similarity (unitary) transformation

then, it is straightforward to check

This shows explicitly that the two representations for J corresponding to j = 1 in (4.72), (4.73) and (4.75) which look rather different are, in fact, related by a similarity transformation and, therefore, are equivalent.

4.3Unitary representations of the Poincaré group

Since we are interested in physical theories which are invariant under translations as well as homogeneous Lorentz transformations, it is useful to study the representations of the Poincaré group. This would help us in understanding the kinds of theories we can consider and the nature of the states they can have. Since Poincaŕe group is non-compact (like the Lorentz group), it is known that it has only infinite dimensional unitary representations except for the trivial representation that is one dimensional. Therefore, we seek to find unitary representations in some infinite dimensional Hilbert space where the generators P_µ, M_µν act as Hermitian operators.

In order to determine the unitary representations, let us note that the operator

defines a quadratic Casimir operator of the Poincaré algebra (4.38) since it commutes with all the ten generators, namely,

The second relation in (4.81) can be intuitively understood as follows. The operators M_µν generate infinitesimal Lorentz transformations through commutation relations and the relation above, which is supposed to characterize the infinitesimal transformation of P², simply implies that P² does not change under a Lorentz transformation (it is a Lorentz scalar) which is to be expected since it does not have any free Lorentz index.

Let us define a new vector operator, known as the Pauli-Lubanski operator, from the generators of the Poincaré group as

The commutator between P_µ and M_νλ introduces metric tensors (see (4.38)) which vanish when contracted with the anti-symmetric Levi-Civita tensor. As a result, the order of P_µ and M_νλ are irrelevant in the definition of the Pauli-Lubanski operator. Furthermore, we note that

which follows from the fact that the generators of translation commute. It follows from (4.83) that, in general, the vector W_µ is orthogonal to P_µ. (However, this is not true for massless theories as we will see shortly.) In general, therefore, (4.83) implies that the Pauli-Lubanski operator has only three independent components (both in the massive and massless cases). Let us define the dual of the generators of Lorentz transformation as

With this, we can write (4.82) also as

where the order of the operators is once again not important.

Let us next calculate the commutators between W^µ and the ten generators of the Poincaré group. First, we have

which follows from the the fact that momenta commute. Consequently, any function of W_µ and, in particular W_µW^µ, will also commute with the generators of translation. We also note that

Here we have used the identity satisfied by the four dimensional Levi-Civita tensors,

Equation (4.87) simply says that under a Lorentz transformation, the operator behaves exactly like the generators of Lorentz transformation (see (4.30)). Namely, it behaves like a second rank anti-symmetric tensor under a Lorentz transformation. Using this, then, we can now evaluate (see (4.37) and (4.87))

In other words, we see that the operator W^µ transforms precisely the same way as does the generator of translation or the P^µ operator under a Lorentz transformation. Namely, it transforms like a vector which we should expect since it has a free Lorentz index. Let us note here, for completeness as well as for later use, that

It follows now from (4.89) that

which is to be expected since W_µW^µ is a Lorentz scalar. Therefore, we conclude that if we define an operator

then, this would also represent a Casimir operator of the Poincaré algebra since W_µ commutes with the generators of translation (see (4.86)). It can be shown that P² and W² represent the only Casimir operators of the algebra and, consequently, the representations can be labelled by the eigenvalues of these operators. In fact, let us note from this analysis that a Casimir operator for the Poincaré algebra must necessarily be a Lorentz scalar (since it has to commute with M_µν). There are other Lorentz scalars that can be constructed from P_µ and M_µν such as

However, it is easy to check that these do not commute with the generators of translation and, therefore, cannot represent Casimir operators of the algebra.

The irreducible representations of the Poincaré group can be classified into two distinct categories, which we treat separately.

4.3.1 Massive representation. To find unitary irreducible representations of the Poincaré algebra, we choose the basis vectors of the representation to be eigenstates of the momentum operators. Namely, without loss of generality, we can choose the momentum operators, P_µ, to be diagonal (they satisfy an Abelian subalgebra). The eigenstates of the momentum operators |p〉 are, of course, labelled by the momentum eigenvalues, p_µ, satisfying

and in this basis, the eigenvalues of the operator P² = P^µP_µ are obvious, namely,

where

Here m denotes the rest mass of the single particle state and we assume the rest mass to be non-zero. However, the eigenvalues of W² are not so obvious. Therefore, let us study this operator in some detail. We recall that

Therefore, using (4.88), we have

where we have simplified terms in the intermediate steps using the anti-symmetry of the Lorentz generators.

To understand the meaning of this operator, let us go to the rest frame of the massive particle. In this frame,

and the operator W² acting on such a state, takes the form

Recalling that (see (4.26))

where J_k represents the total angular momentum of the particle, we obtain

The result in (4.102) can also be derived in an alternative manner which is simpler and quite instructive. Let us note that in the rest frame (4.99), the Pauli-Lubanski operator (4.82) has the form

where we have used (1.34) as well as (4.12). It follows now that

which is the result obtained in (4.102). Therefore, for a massive particle, we can think of W² as being proportional to J² and in the rest frame of the particle, this simply measures the spin of the particle. That is, for a massive particle at rest, we find

Thus, we see that the representations with p² ≠ 0 can be labelled by the eigenvalues (m, s) of the two Casimir operators, namely the mass and the spin of a particle and the dimensionality of such a representation will be (2s + 1) (for both positive as well as negative energy states).

The dimensionality of the representation can also be understood in an alternative manner as follows. For a state at rest with momentum of the form p_µ = (m, 0, 0, 0), we can ask what Lorentz transformations would leave such a vector invariant. Clearly, these would define an invariant subgroup of the Lorentz group and will lead to the degeneracy of states. It is not hard to see that all possible 3-dimensional rotations would leave such a vector invariant. Namely, rotations around the x or the y or the z axis will not change the time component of a four vector (recall that the time component is the spin 0 component of a four vector) and, therefore, would define the stability group of such vectors. Technically, one says that the 3-dimensional rotations define the “little” group of a time-like vector and this method of determining the representation is known as the method of “induced” representation. Therefore, all the degenerate states can be labelled not just by the eigenvalue of the momentum, but also by the eigenvalues of three dimensional rotations, namely, s = 0, 1, · · · and m_s = −s, −s + 1, · · · , s − 1, s. This defines the 2s + 1 dimensional representation for a massive particle of spin s.

This can also be seen algebraically. Namely, a state at rest is an eigenstate of the P₀ operator. From the Lorentz algebra, we note that (see (4.30))

Namely, the operators M_ij, which generate 3-dimensional rotations and are related to the angular momentum operators, commute with P₀. Consequently, the eigenstates of P₀ are invariant under three dimensional rotations and are simultaneous eigenstates of the angular momentum operators as well and such spaces are (2s + 1) dimensional. In closing, let us note from (4.103) that, up to a normalization factor, the three nontrivial Pauli-Lubanski operators correspond to the generators of symmetry of the “little group” in the rest frame.

4.3.2 Massless representation. In contrast to the massive representations of the Poincaré group, the representations for a massless particle are slightly more involved. The basic reason behind this is that the “little” group of a light-like vector is not so obvious. In this case, we note that (we are assuming motion along the z axis and see (4.95))

Consequently, acting on states in such a vector space, we would have (see (4.83))

However, from (4.83) we see that our states in the representation should also satisfy

There now appear two distinct possibilities for the action of the Casimir W² on the states of the representation, namely,

In the first case, namely, for a massless particle if W² ≠ 0, then it can be shown (we will see this at the end of this section) that the representations are infinite dimensional with an infinity of spin values. Such representations do not correspond to physical particles and, consequently, we will not consider such representations.

On the other hand, in the second case where W² = 0 acting on the states of the representation, we can easily show that the action of W^µ in such a space is proportional to that of the momentum operator, namely, acting on states in such a space, W^µ has the form

where h represents a proportionality factor (operator). To determine h, let us recall that

from which it follows that acting on a general momentum basis state |p〉 (not necessarily restricting to massless states), it would lead to (see (4.88))

Comparing with (4.111) we conclude that in this space

This is nothing other than the helicity operator (since L · p = 0) and, therefore, the simultaneous eigenstates of P² and W² would correspond to the eigenstates of momentum and helicity. For completeness, let us note here that in the light-like frame (4.107), the Pauli-Lubanski operator (4.82) takes the form

We see from both (4.103) and (4.115) that the Pauli-Lubanski operator indeed has only three independent components because of the transversality condition (4.83), as we had pointed out earlier. We also note from (4.115) that, in the massless case, W⁰, indeed represents the helicity operator up to a normalization as we had noted in (4.113). It follows now from (4.115) that (the contributions from W⁰ and W³ cancel out)

Let us now determine the dimensionality of the massless representations algebraically. Let us recall that we are considering a massless state with momentum of the form p_µ = (p, 0, 0, −p) and we would like to determine the “little” group of symmetries associated with such a vector. We recognize that in this case, the set of Lorentz transformations which would leave this four vector invariant must include rotations around the z-axis. This can be seen intuitively from the fact that the motion of the particle is along the z axis, but also algebraically by recognizing that a light-like vector of the form being considered is an eigenstate of the operator P₀ − P₃, namely,

Furthermore, from the Poincaré algebra in (4.38), we see that

so that rotations around the z-axis define a symmetry of the light-like vector (state) that we are considering. To determine the other symmetries of a light-like vector, let us define two new operators as

It follows now that these operators commute with P₀ − P₃ in the space of light-like states, namely,

and, therefore, also define symmetries of light-like states. These represent all the symmetries of the light-like vector (state). We note that the algebra of the symmetry generators takes the form

Namely, it is isomorphic to the algebra of the Euclidean group in two dimensions, E₂ (which consists of translations and rotation). Thus, we say that the stability group or the “little” group of a light-like vector is E₂. Clearly, M₁₂ is the generator of rotations around the z axis or in the two dimensional plane and Π₁, Π₂ have the same commutation relations as those of translations in this two dimensional space. Furthermore, comparing with Wⁱ, i = 1, 2, 3 in (4.115), we see that up to a normalization, the three independent Pauli-Lubanski operators are, in fact, the generators of symmetry of the “little” group, as we had also seen in the massive case. This may seem puzzling, but can be easily understood as follows. We note from (4.90) that in the momentum basis states (where p_µ is a number), the Pauli-Lubanski operators satisfy an algebra and, therefore, can be thought of as generators of some transformations. The meaning of the transformations, then, follows from (4.86) as the transformations that leave p_µ invariant. Namely, they generate transformations which will leave the momentum basis states invariant. This is, of course, what we have been investigating within the context of “little” groups.

Let us note from (4.121) that Π₁ ∓ iΠ₂ correspond respectively to raising and lowering operators for M₁₂, namely,

Let us also note for completeness that the Casimir of the E₂ algebra is given by

and comparing with (4.116), we see that in the space of light-like momentum states W² ∝ Π². Since Π₁, Π₂ correspond to generators of “translation”, their eigenvalues can take any value. As a result, if W² ≠ 0 in this space, we note from (4.122) that spin can take an infinite number of values which, as we have already pointed out, does not correspond to any physical system. On the other hand, if W² = 0 in this space of states, then it follows from (4.123) that (h corresponds to the helicity quantum number)

(Alternatively, we can say that Π₁|p, h〉 = 0 = Π₂|p, h〉 and this is the reason for the earlier assertion.) This corresponds to the one dimensional representation of E₂ known as the “degenerate” representation. Clearly, such a state would correspond to the highest or the lowest helicity state. Furthermore, if our theory is also invariant under parity (or space reflection), the space of physical states would also include the state with the opposite helicity (recall that helicity changes sign under space reflection, see (3.148)). As a result, massless theories with nontrivial spin that are parity invariant would have two dimensional representations corresponding to the highest and the lowest helicity states, independent of the spin of the particle. On the other hand, if the theory is not parity invariant, the dimensionality of the representation will be one dimensional, as we have seen explicitly in the case of massless fermion theories describing neutrinos.

Incidentally, the fact that the massless representations have to be one dimensional, in general, can be seen in a heuristic way as follows. Let us consider spin as arising from a circular motion. Then, it is clear that since a massless particle moves at the speed of light, the only consistent circular motion a massless particle can have, is in a plane perpendicular to the direction of motion (otherwise, some component of the velocity would exceed the speed of light). In other words, in such a case, spin can only be either parallel or anti-parallel to the direction of motion leading to the one dimensional nature of the representation. However, if parity (space reflection) is a symmetry of the system, then we must have states corresponding to both the circular motions leading to the two dimensional representation.

4.4References

1.V. Bargmann, Irreducible unitary representations of the Lorentz group, Annals of Mathematics 48, 568 (1947).

2.A. Das and S. Okubo, Lie Groups and Lie Algebras for Physicists, Hindustan Publishing, India and World Scientific Publishing, Singapore (2014).

3.E. Wigner, On unitary representations of the inhomogeneous Lorentz group, Annals of Mathematics 40, 149 (1939).

4.E. Wigner, Group Theory and its Applications to the Quantum Mechanics of Atomic Spectra, Academic Press, New York (1959).

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¹See, for example, Quantum Mechanics: A Modern Introduction, A. Das and A. C. Melissinos (Gordon and Breach), page 289 or Lectures on Quantum Mechanics, A. Das (Hindustan Book Agency, New Delhi), page 182 (note there is a typo in the sign of the 23 element for L₂ in this reference).