Fields transform under the Lorentz group while particle states transform under the Poincaré group.
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Standard quantum field theory literature confirms that relativistic fields transform under representations of the Lorentz group, whereas single-particle states are classified by representations of the Poincaré group.
The unitary irreducible representations of the covering group of the Poincare group define the framework for much of particle physics on the physical Minkowski space = /, where is the Lorentz group. While extraordinarily successful, it does not provide a large enough group of symmetries to encompass observed particles with a (3) classification. Born proposed the reciprocity principle that states physics must be invariant under the reciprocity transform that is heuristically {t, e, qi, pi} → {t, e, pi, −qi} where {t, e, qi, pi} are the time, energy, position and momentum degrees of freedom. This implies that there is reciprocally conjugate relativity principle such that the rates of change of momentum must be bounded by b, where b is a universal constant. The appropriate group of dynamical symmetries that embodies this is the canonical group (1, 3) = (1, 3) ⊗s (1, 3) = (1, 3) ⊗s (1, 3) and in this theory the non-commuting space = (1, 3)/(1, 3) is the physical quantum space endowed with a metric that is the second Casimir invariant of the canonical group, T2 + E2/c2b2 − Q2/c2 − P2/b2 + 2I/bc (Y/bc − 2) where {T, E, Qi, Pi, I, Y} are the generators of the algebra of (1, 3) = (1) ⊗s (1, 3). The idea is to study the representations of the canonical dynamical group using Mackey's theory to determine whether the representations can encompass the spectrum of particle states. The unitary irreducible representations of the canonical group contain a direct product term that is a representation of (1, 3) that Kalman has studied as a dynamical group for hadrons. The (1, 3) representations contain discrete series that may be decomposed into infinite ladders where the rungs are representations of (3) (finite dimensional) or (2) (with degenerate (1) ⊗ (2) finite-dimensional representations) corresponding to the rest or null frames.
In the quantum theory of fields one writes the relativistic field operator as a linear combination of annihilation operators, with invariant coefficient functions. The annihilation operators transform as physical, massive, single particle states with a unitary representation of the Poincare group, while the relativistic field operator transforms with a nonunitary spin 1/2 representation of the homogeneous Lorentz group. The Lorentz group represents translations trivially, i.e. as multipliction by unity. Here the nonunitary representation is provided with translation matrices, so that the unitary and the nonunitary representations represent the same group, the Poincare group. Translation matrix invariance is shown to give the free particle Dirac equation, without invoking parity. The coefficient functions for a given momentum determine a current. These currents turn out to be, within a constant factor, the electromagnetic vector potential of the free particle source moving with that momentum. Thus it is shown that the Dirac and Maxwell equations can be related to the inclusion of translation matrices in the transformations of field operators.
In standard quantum field theory, the one-particle states are classified by the unitary representations of the Poincar\'e group, whereas the causal fields' classification employs the finite-dimensional (non-unitary) representations of the (homogeneous) Lorentz group. We investigate the possibility of constructing fields that transform under the full representation of the Poincar\'e group. We show that such fields can be consistently constructed, although the Lagrangians that describe them exhibit explicit dependence on the space-time coordinates. The inclusion of gravity within the framework of the Poincar\'e gauge theory is then discussed. A new feature that occurs is that the translational gauge fields enter the covariant derivative of matter fields. The Poincar\'e-gauge approach works still well and leads to interesting consequences. The detailed discussion of the Dirac field is presented and the relation to the earlier accounts on Poincar\'e-spinors is drawn. Another example that is considered is the Poincar\'e-vector field. The presentation has a partly didactic character and is addressed to all the readers who are interested in the rudiments of quantum field theory and the gauge description of gravity.
This book discusses how relativistic quantum field theories must transform under strongly continuous unitary representations of the Poincaré group. The focus is on the construction of the representations that provide the basis for the formulation of current relativistic quantum field theories of scalar fields, the Dirac field, and the electromagnetic field. Such construction is tied to the use of the methods of operator theory that also provide the basis for the formulation of quantum mechanics, up to the interpretation of the measurement process. In addition, since representation spaces of primary interest in quantum theory are infinite dimensional, the use of these methods is essential. Consequently, the book also calculates the generators of relevant strongly continuous one-parameter groups that are associated with the representations and, where appropriate, the corresponding spectrum. Part I of Quantum Spin and Representations of the Poincaré Group specifically addresses: conventions; basic properties of SO(2) and SO(3); construction of a double cover of SO(3); SU(2) spinors; continuous unitary representation of SU(2); basic properties of the Lorentz Group; unitary representation of the restricted Lorentz Group; an extension to a strongly continuous representation of the restricted Poincaré Group; and an extension to a unitary/anti-unitary representation of the Poincaré Group.
Dirac algebra Gamma matrices Lorentz group Möbius transformation Poincaré group Representation theory of the Poincaré group Symmetry in quantum mechanics
The Lorentz group is a Lie group of symmetries of the spacetime of special relativity. This group can be realized as a collection of matrices, linear transformations, or unitary operators on some Hilbert space; it has a variety of representations. This group is significant because special relativity together with quantum mechanics are the two physical theories that are most thoroughly establishe
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and similarly for the annihilation operator. The point to be made is that the field operator transforms according to a finite-dimensional non-unitary representation of the Lorentz group, while the creation operator transforms under the infinite-dimensional unitary representation of the Poincare group characterized by the mass and spin (m, s) of the particle. The connection between the two are the wave functions, also called coefficient functions
where D is the non-unitary Lorentz group representative of Λ and D(s) is a unitary representative of the so-called Wigner rotation R associated to Λ and p that derives from the representation of the Poincaré group, and s is the spin of the particle.
All of the above formulas, including the definition of the field operator in terms of creation and annihilation operators, as well as the differential equations satisfied by the field operator for a particle with specified mass, spin and the (m, n) representation under which it is supposed to transform, and also that of the wave function, can be derived from group theoretical considerations alone once the frameworks of quantum mechanics and special relativity is given.
Representations of the Canonical group, (the semi-direct product of the Unitary and Weyl-Heisenberg groups), acting as a dynamical group on noncommuting extended phase space
The unitary irreducible representations of the covering group of the Poincare group P define the framework for much of particle physics on the physical Minkowski space P/L, where L is the Lorentz group. While extraordinarily successful, it does not provide a large enough group of symmetries to encompass observed particles with a SU(3) classification. Born proposed the reciprocity principle that states physics must be invariant under the reciprocity transform that is heuristically {t,e,q,p}->{t,e,p,-q} where {t,e,q,p} are the time, energy, position, and momentum degrees of freedom. This implies that there is reciprocally conjugate relativity principle such that the rates of change of momentum must be bounded by b, where b is a universal constant.
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