Lectures on Quantum Field Theory. Ashok Das

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