Only when our Universe's last two, presumably rather large Black holes have collided and the last pop of Hawking radiation and gravitational waves is emanated as maximal Thermodynamic Entropy is reached will the universe's Chirality be one and the same as its Helicity, massless as it will be once again.
The stress-energy tensor is the source of space-time curvature and can be viewed as 4x4 matrix. The time–time component is the density of relativistic mass, e.g. the energy density. The time-space component is the density of the i-th component of Linear momentum or (relativistic) mass per unit volume.
From the diagram we see that Linear momentum is a vector invariant under translations. It arises from the assumed homogeneity of space: you move your experiment through space and unless the environment is permeated by markedly different fields or objects (!) it will deliver the same observations. Energy is the scalar invariant arising from the invariance of closed systems to translations in time. That is Energy is the conserved (Noether) current as time is deemed homogenous and indiscernibly the same so that an experiment done today then tomorrow (with all other things being equal) will deliver the same results .
Linear momentum and energy are formed by contractions of the energy-momentum tensor with the Killing field generator of their isometries. In the Lie vector field sense the local metric is preserved when Lie dragged along these tangent vector fields.
Similarly Angular momentum arises from the isotropy of space, it is a pseudo-vector (since it has a handedness -directionality) whereas rotational kinetic energy is a pseudo-scalar. Neither is elaborated on the diagram. These would arise from the spatial diagonal elements of the stress-energy tensor that represent normal stress or "pressure" and the off diagonal spatial components that are the shear stress.
Discrete Symmetries in field theories
In quantum field theories we assume fields exists in a exogenously defined flat space-time imbued with the symmetries of Poincare. At a high level we come across the following terminology:1- "Handedness" is the "Parity", P of CPT.
2- Charge Parity, CP invariance is broken through Weak interaction decay channels so are not deemed exact symmetries.
3-CPT is deemed sacrosanct so that Poincare (Lorenz plus space-time translation) invariance is preserved.
As such broken CP implies Time (inversion), T symmetry is broken a problem given the even more sacrosanct Zeroth Law of Thermodynamics.
Field theorists distinguish two possible particle states:
The latter is defined via the direction of the momentum and angular momentum of a particle and is not a Lorentz-invariant property for massive particles (as you can attached an inertial frame to them). In the massless case (of photons, possibly gravitons and neutrinos) however, helicity can be linked directly to chirality.
As such we talk of two-spinor, (self dual) "chiral formulations of gravity, with their associated (linearised) quantised spin 2- helicity graviton states. Gravitational as well as electromagnetic waves were subject to polarisation at recombination and are imprinted on the CMBR. Traditionally neutrinos are represented by chiral Weyl-Spinors whether or not they are massless and subject to some mixing theorem.
The particulates that give rise to the interaction between massive particles are the fields of force mediating bosons of Quantum Electrodynamics (QED), Quantum Chromodynamics and some quantum field theory of gravity. They are respectively the photon, gluon, and graviton each possessing an unambiguous chirality (‘handedness”) that is the same as their helicity (‘twistedness”).
These massless particles have spins that are in the same direction along its axis of motion regardless of the point of view of the observer. That their Chirality is absolute in this sense is due to both the invariance and finite speed of light. That such massless particles move at the speed of light, means that a massive observer (travelling at less than the speed of light) cannot travel in a faster reference frame in which the particle would appear to reverse its relative direction.
Their handedness is unambiguous in that all real observers see the same chirality. Accordingly we say that the direction of spin of the massless particles is not affected by a Lorentz boost (the relativistic equivalent of a Galilean change of reference frame) in the direction of motion of the particle. As such the sign of the projection (helicity) is fixed for all reference frames.
A photon’s “twistedness” has a sense in that helix described the rotation of its electric field (say) can be clockwise or anti-clockwise. By definition the helicity of a particle is right-handed if the direction of its spin is the same as the direction of its motion. It is left-handed if the directions of spin and motion are opposite.
Mathematically, helicity is the sign of the projection of the spin vector onto the momentum vector: left is negative, right is positive. This can be cast in the language of linear operators acting on a vector space. For our purposes think of a matrix acting on a column vector.
Helicity is always rotationally invariant but for massless particles it is also Lorentz invariant: no boosts are possible that will reverse the sign of the 3-momentum, p of the particle. The sign of p though does reverse under a Parity transformation as although the momentum is a true (polar) vector changing sign with x, angular momentum vector, J is a pseudo-vector (axial vector) remaining unchanged with a reversal of x. Helicity which is the scalar product of the two J.p is thus a pseudo scalar changing sign under a parity transformation. In order to preserve parity both potential helictites of the particle need to be present and this is the case for the photon with spin ±1.
The spin, s of a massive particle means something different to the spin of a massless particle. We view the generators of the Proper Lorentz group as Angular Momentum (pseudo) vectors. They are generators, L in the sense that they infinitesimally create boosts by the exponential operation exp(𝟄L) between the algebra and group - think roughly here of a Taylor expansion of X around 0 for a small epsilon, 𝟄. They are (vector) operators in the sense that they operate linearly in a linear space so that compositions of them are additive. But they are special vectors being axial in nature with a definite orientation.a ll this means that they are closed under the [,] composition law of commutation:[J,J]=ieJ
- "chirality" - from the Greek, χειρ, ’hand’
- "helicity"-from the Greek, ελιξ, ’twisted’
The latter is defined via the direction of the momentum and angular momentum of a particle and is not a Lorentz-invariant property for massive particles (as you can attached an inertial frame to them). In the massless case (of photons, possibly gravitons and neutrinos) however, helicity can be linked directly to chirality. As such we talk of two-spinor, (self dual) "chiral formulations of gravity, with their associated (linearised) quantised spin 2- helicity graviton states. Gravitational as well as electromagnetic waves were subject to polarisation at recombination and are imprinted on the CMBR. Traditionally neutrinos are represented by chiral Weyl-Spinors whether or not they are massless and subject to some mixing theorem.
The Relativity of Helicity
The particulates that give rise to the interaction between massive particles are the fields of force mediating bosons of Quantum Electrodynamics (QED), Quantum Chromodynamics and some quantum field theory of gravity. They are respectively the photon, gluon, and graviton each possessing an unambiguous chirality (‘handedness”) that is the same as their helicity (‘twistedness”).
These massless particles have spins that are in the same direction along its axis of motion regardless of the point of view of the observer. That their Chirality is absolute in this sense is due to both the invariance and finite speed of light. That such massless particles move at the speed of light, means that a massive observer (travelling at less than the speed of light) cannot travel in a faster reference frame in which the particle would appear to reverse its relative direction.
Their handedness is unambiguous in that all real observers see the same chirality. Accordingly we say that the direction of spin of the massless particles is not affected by a Lorentz boost (the relativistic equivalent of a Galilean change of reference frame) in the direction of motion of the particle. As such the sign of the projection (helicity) is fixed for all reference frames.
A photon’s “twistedness” has a sense in that helix described the rotation of its electric field (say) can be clockwise or anti-clockwise. By definition the helicity of a particle is right-handed if the direction of its spin is the same as the direction of its motion. It is left-handed if the directions of spin and motion are opposite.
Poincare Algebra of Helicity states
Mathematically, helicity is the sign of the projection of the spin vector onto the momentum vector: left is negative, right is positive. This can be cast in the language of linear operators acting on a vector space. For our purposes think of a matrix acting on a column vector.
Helicity is always rotationally invariant but for massless particles it is also Lorentz invariant: no boosts are possible that will reverse the sign of the 3-momentum, p of the particle. The sign of p though does reverse under a Parity transformation as although the momentum is a true (polar) vector changing sign with x, angular momentum vector, J is a pseudo-vector (axial vector) remaining unchanged with a reversal of x. Helicity which is the scalar product of the two J.p is thus a pseudo scalar changing sign under a parity transformation. In order to preserve parity both potential helictites of the particle need to be present and this is the case for the photon with spin ±1.
The spin, s of a massive particle means something different to the spin of a massless particle. We view the generators of the Proper Lorentz group as Angular Momentum (pseudo) vectors. They are generators, L in the sense that they infinitesimally create boosts by the exponential operation exp(𝟄L) between the algebra and group - think roughly here of a Taylor expansion of X around 0 for a small epsilon, 𝟄. They are (vector) operators in the sense that they operate linearly in a linear space so that compositions of them are additive. But they are special vectors being axial in nature with a definite orientation.a ll this means that they are closed under the [,] composition law of commutation:[J,J]=ieJ
In addition to the boost invariance we have considered so far, so-called Poincare invariance includes symmetry under translations of space-time. That is if we shift space or time incrementally along (generated by a vector, P) by say redefining the zero point of our co-ordinate axis we observe that our laws of motion do not change.
An operator that has two eigenvalues ±1 defines the Chirality for a massive Dirac fermion. Any Dirac field can therefore be projected into its left- or right-handed (chiral) Weyl components. The coupling of the weak interaction to fermions is proportional to such a projection operator, which is responsible for its parity symmetry violation.
What we look for in analysing the algebraic structure is a combination, “C" of operators (J and P), which commute with all the other generators. If that is the case we can say that their are states in the linear system that are simultaneous eigenstates of C(J,P). For the Poincare group there are two such "Casimir" combinations and as such a simultaneous state possesses two eigenvalues- one of mass and one of spin.
For massive particles it turns out that the sub-group of Lorentz boosts that leave 4-momentum untouched induces a representation of the Poincare group. This sub-group has the character of 3-dimensional rotations and is covered by the (symplectic) spin group.
It turns out then that the set of generators J and P are closed under [,] composition law and thus particles have two qualities: a definite mass and a spin. The mass is associated to the 4-momentum, P the spin to J. So in the massive case we have a SU(2) representation in which the Casimir operator J^2 has eigenvalues s(s+1) and J_3 values of s and -s. In the massless case only the angular momentum operator is specified by its Helicity. For a neutrino a massless fermion produced in beta decay the spin is -1/2.
Nature appears to treat (Charge Parity and Time) CPT invariance as a fundamental symmetry such that particles linked by them in unison are deemed indistinguishable from each other. CP transformations do however distinguish certain meson decays which express an orientation preference.
Three possible symmetries to consider as invariants of nature, which in fact only usually, but not always, hold, are those of charge conjugation (C), parity (P), and time reversal (T):
Charge conjugation(C): reversing the electric charge and all the internal quantum number
Parity (P): space inversion; reversal of the space coordinates, but not the time
Time reversal (T): replacing t by -t. This reverses time derivatives like momentum and angular momentum.
It is a reasonable presupposition that nature should not care whether its coordinate system is right-handed or left-handed, but surprisingly, that turns out not to be so. In a famous experiment by C. S. Wu, the non-conservation of parity in beta decay was demonstrated.
This and subsequent experiments have consistently shown that a neutrino always has its intrinsic angular momentum (spin) pointed in the direction opposite its velocity. It is called a left-handed particle as a result. Anti-neutrinos have their spins parallel to their velocity and are therefore right-handed particles. Therefore we say that the neutrino has an intrinsic parity.
Nature at a very fundamental level distinguishes "left-handed" and "right-handed" systems. The combination of the parity operation (=P) and "charge conjugation" (changing each particle into its antiparticle = C) thought initially to be an inviolate conservation law (CP invariance), through the study of Kaon decay in 1964 was shown to be only a partial law. Only upon adding time reversal (=T) to our arsenal of transformations will a system be deemed indistinguishable from its original state.
One could ask whether CPT invariance implied Lorentz or whether invariance under Lorentz transformations implies CPT invariance. Either way it is the latter that governs the following observations:
The notion of the chirality of a particle is both clarified and made more abstract by studying all the (Group of) possible transformations that do not materially affect the representative object in either real or mathematical abstract spin space.
The Chirality of particle is determined by whether the particle object transforms as either right or left-handed representations of the Poincare group: a connected set of orientation preserving Lorentz boosts and rotations with the addition of translational invariance. We note that different representative mathematical objects, despite being rooted from a common hierarchy, can represent certain particles consistently. Some representations, such as Dirac spinors used to describe massive fermions (like electrons), have both right and left-handed components, others such as the Weyl-Spinor describing (essentially) massless neutrinos have a handedness.
For massive fermion (spin half) particles (in which Lorentz boosts can operate at positive relative velocities) such as electrons, quarks, and neutrinos—chirality and helicity must be distinguished. As it is possible for an observer to change to a reference frame that overtakes these spinning particles, the particle will appear to move backwards, and its helicity (now only an 'apparent chirality') will be reversed. For massive particles therefore chirality is not the same as helicity.
To finish a little let us review the structure of the full Lorentz group, O(1,3). It is relevant for the CPT theorem and has that has four disconnected, disjoint components according to the signs of the determinant of the Lorentz transformation, det(Λ)=±1.
For sets of boosts such that Λtt=γ>1, called "orthochronous proper(linear) transformations we form the restricted sub group SO+(1,3). When you see determinants, think of oriented parallel-pipeds, so this sub-group has already been quotiented out by P.
In increasing generality we have then the:
The lowest-dimensional (non-trivial irreducible) representations of the proper orthochronous Lorentz group, are the relativistically generalized Pauli matrices.
The special linear group SL(2, C) in two complex dimensions is defined by:
SL(2, C) = {A∈GL(2, C)| detA = +1}. Here the SU(2) matrices (of Pauli) in the SL(2, C) generate spatial rotations; Hermitian matrices in the SL(2, C) generate the boosts.
Denoting the "fundamental" representation of the spinor Lorentz group SL(2, C) (by itself) as the (1/2, 0) representation and its complex conjugate, (1/2, 0)∗ by (0,1/2) we have: left "chiral" spinors: "chirality- Weyl spinors" that transform as (0,1/2)=(1/2, 0)∗
In most circumstances, two left-handed fermions interact more strongly than right-handed or opposite-handed fermions implying that the universe has a preference for left-handed chirality, which violates symmetry of the other forces of nature. Indeed only left-handed fermions interact with the weak interaction. There is no frame dependence of the weak interaction: a particle that interacts with the weak force does so in every frame.
Casimir Operator
An operator that has two eigenvalues ±1 defines the Chirality for a massive Dirac fermion. Any Dirac field can therefore be projected into its left- or right-handed (chiral) Weyl components. The coupling of the weak interaction to fermions is proportional to such a projection operator, which is responsible for its parity symmetry violation.
What we look for in analysing the algebraic structure is a combination, “C" of operators (J and P), which commute with all the other generators. If that is the case we can say that their are states in the linear system that are simultaneous eigenstates of C(J,P). For the Poincare group there are two such "Casimir" combinations and as such a simultaneous state possesses two eigenvalues- one of mass and one of spin.
For massive particles it turns out that the sub-group of Lorentz boosts that leave 4-momentum untouched induces a representation of the Poincare group. This sub-group has the character of 3-dimensional rotations and is covered by the (symplectic) spin group.
It turns out then that the set of generators J and P are closed under [,] composition law and thus particles have two qualities: a definite mass and a spin. The mass is associated to the 4-momentum, P the spin to J. So in the massive case we have a SU(2) representation in which the Casimir operator J^2 has eigenvalues s(s+1) and J_3 values of s and -s. In the massless case only the angular momentum operator is specified by its Helicity. For a neutrino a massless fermion produced in beta decay the spin is -1/2.
Nature appears to treat (Charge Parity and Time) CPT invariance as a fundamental symmetry such that particles linked by them in unison are deemed indistinguishable from each other. CP transformations do however distinguish certain meson decays which express an orientation preference.
CPT Invariance
Three possible symmetries to consider as invariants of nature, which in fact only usually, but not always, hold, are those of charge conjugation (C), parity (P), and time reversal (T):
Charge conjugation(C): reversing the electric charge and all the internal quantum number
Parity (P): space inversion; reversal of the space coordinates, but not the time
Time reversal (T): replacing t by -t. This reverses time derivatives like momentum and angular momentum.
It is a reasonable presupposition that nature should not care whether its coordinate system is right-handed or left-handed, but surprisingly, that turns out not to be so. In a famous experiment by C. S. Wu, the non-conservation of parity in beta decay was demonstrated.
This and subsequent experiments have consistently shown that a neutrino always has its intrinsic angular momentum (spin) pointed in the direction opposite its velocity. It is called a left-handed particle as a result. Anti-neutrinos have their spins parallel to their velocity and are therefore right-handed particles. Therefore we say that the neutrino has an intrinsic parity.
Nature at a very fundamental level distinguishes "left-handed" and "right-handed" systems. The combination of the parity operation (=P) and "charge conjugation" (changing each particle into its antiparticle = C) thought initially to be an inviolate conservation law (CP invariance), through the study of Kaon decay in 1964 was shown to be only a partial law. Only upon adding time reversal (=T) to our arsenal of transformations will a system be deemed indistinguishable from its original state.
One could ask whether CPT invariance implied Lorentz or whether invariance under Lorentz transformations implies CPT invariance. Either way it is the latter that governs the following observations:
- Integer spin particles obey Bose-Einstein statistics and half-integer spin particles obey Fermi-Dirac statistics. Operators with integer spins must be quantized using commutation relations, while anti-commutation relations must be used for operators with half integer spin.
- That particles and antiparticles have identical masses and their lifetimes arises from CPT invariance of physical theories.
- All the internal quantum numbers of antiparticles are opposite to those of the particles.
Dirac Spinors
The notion of the chirality of a particle is both clarified and made more abstract by studying all the (Group of) possible transformations that do not materially affect the representative object in either real or mathematical abstract spin space.
The Chirality of particle is determined by whether the particle object transforms as either right or left-handed representations of the Poincare group: a connected set of orientation preserving Lorentz boosts and rotations with the addition of translational invariance. We note that different representative mathematical objects, despite being rooted from a common hierarchy, can represent certain particles consistently. Some representations, such as Dirac spinors used to describe massive fermions (like electrons), have both right and left-handed components, others such as the Weyl-Spinor describing (essentially) massless neutrinos have a handedness.
For massive fermion (spin half) particles (in which Lorentz boosts can operate at positive relative velocities) such as electrons, quarks, and neutrinos—chirality and helicity must be distinguished. As it is possible for an observer to change to a reference frame that overtakes these spinning particles, the particle will appear to move backwards, and its helicity (now only an 'apparent chirality') will be reversed. For massive particles therefore chirality is not the same as helicity.
Proper Orthochronous Lorentz Group
To finish a little let us review the structure of the full Lorentz group, O(1,3). It is relevant for the CPT theorem and has that has four disconnected, disjoint components according to the signs of the determinant of the Lorentz transformation, det(Λ)=±1.
For sets of boosts such that Λtt=γ>1, called "orthochronous proper(linear) transformations we form the restricted sub group SO+(1,3). When you see determinants, think of oriented parallel-pipeds, so this sub-group has already been quotiented out by P.
In increasing generality we have then the:
- Proper orthochronous Lorentz group, SO+(1,3) considered to be the true (local) symmetry group of all physical laws governing quantum field theories in classical spacetimes.
- Special Orthogonal, SO(1, 3) group that includes spacetime reflections (PT) with γ ≤ −1,
- Full O(1, 3) group that contains parity (P) and time reversal (T) transformations not related to exact symmetries of the real world.
The lowest-dimensional (non-trivial irreducible) representations of the proper orthochronous Lorentz group, are the relativistically generalized Pauli matrices.
The special linear group SL(2, C) in two complex dimensions is defined by:
SL(2, C) = {A∈GL(2, C)| detA = +1}. Here the SU(2) matrices (of Pauli) in the SL(2, C) generate spatial rotations; Hermitian matrices in the SL(2, C) generate the boosts.
Denoting the "fundamental" representation of the spinor Lorentz group SL(2, C) (by itself) as the (1/2, 0) representation and its complex conjugate, (1/2, 0)∗ by (0,1/2) we have: left "chiral" spinors: "chirality- Weyl spinors" that transform as (0,1/2)=(1/2, 0)∗
In most circumstances, two left-handed fermions interact more strongly than right-handed or opposite-handed fermions implying that the universe has a preference for left-handed chirality, which violates symmetry of the other forces of nature. Indeed only left-handed fermions interact with the weak interaction. There is no frame dependence of the weak interaction: a particle that interacts with the weak force does so in every frame.
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