Chirality (physics)

A chiral phenomenon is one that is not identical to its mirror image (see the article on mathematical chirality). The spin of a particle may be used to define a handedness, or helicity, for that particle, which, in the case of a massless particle, is the same as chirality. A symmetry transformation between the two is called parity transformation. Invariance under parity transformation by a Dirac fermion is called chiral symmetry.

Chirality and helicity

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. So a standard clock, with its spin vector defined by the rotation of its hands, tossed with its face directed forwards, has left-handed helicity.

Mathematically, helicity is the sign of the projection of the spin vector onto the momentum vector: “left” is negative, “right” is positive.

Right left helicity

The chirality of a particle is more abstract: It is determined by whether the particle transforms in a right- or left-handed representation of the Poincaré group.[a]

For massless particles – photons, gluons, and (hypothetical) gravitons – chirality is the same as helicity; a given massless particle appears to spin in the same direction along its axis of motion regardless of point of view of the observer.

For massive particles – such as electrons, quarks, and neutrinos – chirality and helicity must be distinguished: In the case of these particles, it is possible for an observer to change to a reference frame moving faster than the spinning particle, in which case the particle will then appear to move backwards, and its helicity (which may be thought of as “apparent chirality”) will be reversed.

A massless particle moves with the speed of light, so no real observer (who must always travel at less than the speed of light) can be in any reference frame where the particle appears to reverse its relative direction of spin, meaning that all real observers see the same helicity. Because of this, the direction of spin of massless particles is not affected by a change of viewpoint (Lorentz boost) in the direction of motion of the particle, and the sign of the projection (helicity) is fixed for all reference frames: The helicity of massless particles is a “relativistic invariant” (a quantity whose value is the same in all inertial reference frames) which always matches the massless particles' chirality.

The discovery of neutrino oscillation implies that neutrinos have mass, so the only observed massless particle is the photon. The gluon is also expected to be massless, although the assumption that it is has not been conclusively tested. Hence, these are the only two particles now known for which helicity could be identical to chirality, and only one of them has been confirmed by measurement. All other observed particles have mass and thus may have different helicities in different reference frames. It is still possible that as-yet unobserved particles, like the graviton, might be massless, and hence have invariant helicity that matches their chirality, like the photon.

Chiral theories

Only left-handed fermions and right-handed antifermions interact with the weak interaction.[1] 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 a symmetry of the other forces of nature.

Chirality for a Dirac fermion ψ is defined through the operator γ5, which has eigenvalues ±1. Any Dirac field can thus be projected into its left- or right-handed component by acting with the projection operators ½(1−γ5) or ½(1+γ5) on ψ.

The coupling of the charged weak interaction to fermions is proportional to the first projection operator, which is responsible for this interaction's parity symmetry violation.

A common source of confusion is due to conflating this operator with the helicity operator. Since the helicity of massive particles is frame-dependent, it might seem that the same particle would interact with the weak force according to one frame of reference, but not another. The resolution to this false paradox is that the chirality operator is equivalent to helicity for massless fields only, for which helicity is not frame-dependent. By contrast, for massive particles, chirality is not the same as helicity, so there is no frame dependence of the weak interaction: a particle that couples to the weak force in one frame does so in every frame.

A theory that is asymmetric with respect to chiralities is called a chiral theory, while a non-chiral (i.e., parity-symmetric) theory is sometimes called a vector theory. Many pieces of the Standard Model of physics are non-chiral, which is traceable to anomaly cancellation in chiral theories. Quantum chromodynamics is an example of a vector theory, since both chiralities of all quarks appear in the theory, and couple to gluons in the same way.

The electroweak theory, developed in the mid 20th century, is an example of a chiral theory. Originally, it assumed that neutrinos were massless, and only assumed the existence of left-handed neutrinos (along with their complementary right-handed antineutrinos). After the observation of neutrino oscillations, which imply that neutrinos are massive like all other fermions, the revised theories of the electroweak interaction now include both right- and left-handed neutrinos. However, it is still a chiral theory, as it does not respect parity symmetry.

The exact nature of the neutrino is still unsettled and so the electroweak theories that have been proposed are somewhat different, but most accommodate the chirality of neutrinos in the same way as was already done for all other fermions.

Chiral symmetry

Vector gauge theories with massless Dirac fermion fields ψ exhibit chiral symmetry, i.e., rotating the left-handed and the right-handed components independently makes no difference to the theory. We can write this as the action of rotation on the fields:




With N flavors, we have unitary rotations instead: U(N)L×U(N)R.

More generally, we write the right-handed and left-handed states as a projection operator acting on a spinor. The right-handed and left-handed projection operators are


Massive fermions do not exhibit chiral symmetry, as the mass term in the Lagrangian, mψ, breaks chiral symmetry explicitly.

Spontaneous chiral symmetry breaking may also occur in some theories, as it most notably does in quantum chromodynamics.

The chiral symmetry transformation can be divided into a component that treats the left-handed and the right-handed parts equally, known as vector symmetry, and a component that actually treats them differently, known as axial symmetry.[2] (cf. Current algebra.) A scalar field model encoding chiral symmetry and its breaking is the chiral model.

The most common application is expressed as equal treatment of clockwise and counter-clockwise rotations from a fixed frame of reference.

The general principle is often referred to by the name chiral symmetry. The rule is absolutely valid in the classical mechanics of Newton and Einstein, but results from quantum mechanical experiments show a difference in the behavior of left-chiral versus right-chiral subatomic particles.

Example: u and d quarks in QCD

Consider quantum chromodynamics (QCD) with two massless quarks u and d (massive fermions do not exhibit chiral symmetry). The Lagrangian reads

In terms of left-handed and right-handed spinors, it reads

(Here, i is the imaginary unit and the Dirac operator.)


it can be written as

The Lagrangian is unchanged under a rotation of qL by any 2×2 unitary matrix L, and qR by any 2×2 unitary matrix R.

This symmetry of the Lagrangian is called flavor chiral symmetry, and denoted as U(2)L×U(2)R. It decomposes into

The singlet vector symmetry, U(1)V, acts as

and corresponds to baryon number conservation.

The singlet axial group U(1)A acts as

and it does not correspond to a conserved quantity, because it is explicitly violated by a quantum anomaly.

The remaining chiral symmetry SU(2)L×SU(2)R turns out to be spontaneously broken by a quark condensate formed through nonperturbative action of QCD gluons, into the diagonal vector subgroup SU(2)V known as isospin. The Goldstone bosons corresponding to the three broken generators are the three pions. As a consequence, the effective theory of QCD bound states like the baryons, must now include mass terms for them, ostensibly disallowed by unbroken chiral symmetry. Thus, this chiral symmetry breaking induces the bulk of hadron masses, such as those for the nucleons — in effect, the bulk of the mass of all visible matter.

In the real world, because of the nonvanishing and differing masses of the quarks, SU(2)L×SU(2)R is only an approximate symmetry[3] to begin with, and therefore the pions are not massless, but have small masses: they are pseudo-Goldstone bosons.[4]

More Flavors

For more "light" quark species, N flavors in general, the corresponding chiral symmetries are U(N)L×U(N)R, decomposing into

and exhibiting a very analogous chiral symmetry breaking pattern.

Most usually, N = 3 is taken, the u, d, and s quarks taken to be light (the Eightfold way (physics)), so then approximately massless for the symmetry to be meaningful to a lowest order, while the other three quarks are sufficiently heavy to barely have a residual chiral symmetry be visible for practical purposes.

An application in Particle Physics

In theoretical physics, the electroweak model breaks parity maximally. All its fermions are chiral Weyl fermions, which means that the charged weak gauge bosons only couple to left-handed quarks and leptons. (Note that the neutral electroweak Z boson couples to left and right-handed fermions.)

Some theorists found this objectionable, and so conjectured a GUT extension of the weak force which has new, high energy W' and Z' bosons, which now couple with right handed quarks and leptons:


Here, SU(2)L (pronounced “SU(2) left”) is none other than SU(2)W from above, while B−L is the baryon number minus the lepton number. The electric charge formula in this model is given by


where and are the left and right weak isospin values of the fields in the theory.

There is also the chromodynamic SU(3)C. The idea was to restore parity by introducing a left-right symmetry. This is a group extension of Z2 (the left-right symmetry) by

to the semidirect product

This has two connected components where Z2 acts as an automorphism, which is the composition of an involutive outer automorphism of SU(3)C with the interchange of the left and right copies of SU(2) with the reversal of U(1)B−L. It was shown by Rabindra N. Mohapatra and Goran Senjanovic in 1975 that left-right symmetry can be spontaneously broken to give a chiral low energy theory, which is the Standard Model of Glashow, Weinberg, and Salam and it also connects the small observed neutrino masses to the breaking of left-right symmetry via the seesaw mechanism.

In this setting, the chiral quarks


are unified into an irreducible representation (“irrep”)

The leptons are also unified into an irreducible representation

The Higgs bosons needed to implement the breaking of left-right symmetry down to the Standard Model are

This then predicts three sterile neutrinos, which is perfectly consistent with current neutrino oscillation data. Within the seesaw mechanism, the sterile neutrinos become superheavy without affecting physics at low energies.

Because the left-right symmetry is spontaneously broken, left-right models predict domain walls. This left-right symmetry idea first appeared in the Pati–Salam model (1974), Mohapatra–Pati models (1975).

See also


  1. ^ Note, however, that representations such as Dirac spinors and others, have both right- and left-handed components. In cases like this, we can define projection operators that project out either the right or left hand components and discuss the right- and left-handed portions of the representation.


  1. ^ Bogdan Povh, Klaus Rith, Christoph Scholz, Frank Zetsche, Particles and Nuclei: An Introduction to the Physical Concepts, (Springer 2006) p. 145, ISBN 978-3-540-36683-6
  2. ^ Ta-Pei Cheng and Ling-Fong Li, Gauge Theory of Elementary Particle Physics, (Oxford 1984) ISBN 978-0198519614
  3. ^ Gell-Mann, M.; Renner, B. (1968). "Behavior of Current Divergences under SU_{3}×SU_{3}". Physical Review. 175 (5): 2195. Bibcode:1968PhRv..175.2195G. doi:10.1103/PhysRev.175.2195.
  4. ^ Peskin, Michael; Schroeder, Daniel (1995). An Introduction to Quantum Field Theory. Westview Press. p. 670. ISBN 0-201-50397-2.
  • Walter Greiner and Berndt Müller (2000). Gauge Theory of Weak Interactions. Springer. ISBN 3-540-67672-4.CS1 maint: Uses authors parameter (link)
  • Gordon L. Kane (1987). Modern Elementary Particle Physics. Perseus Books. ISBN 0-201-11749-5.
  • Kondepudi, Dilip K.; Hegstrom, Roger A. (January 1990). "The Handedness of the Universe". Scientific American. 262 (1): 108–115.
  • Winters, Jeffrey (November 1995). "Looking for the Right Hand". Discover. Retrieved 12 September 2015.

External links

Chirality (chemistry)

Chirality is a geometric property of some molecules and ions. A chiral molecule/ion is non-superposable on its mirror image. The presence of an asymmetric carbon center is one of several structural features that induce chirality in organic and inorganic molecules. The term chirality is derived from the Ancient Greek word for hand, χεῖρ (kheir).

The mirror images of a chiral molecule or ion are called enantiomers or optical isomers. Individual enantiomers are often designated as either right-handed or left-handed. Chirality is an essential consideration when discussing the stereochemistry in organic and inorganic chemistry. The concept is of great practical importance because most biomolecules and pharmaceuticals are chiral.

Chiral molecules and ions are described by various ways of designating their absolute configuration, which codify either the entity's geometry or its ability to rotate plane-polarized light, a common technique in studying chirality.

Chirality (disambiguation)

Chirality (handedness) is a property of asymmetry.

Chirality may also refer to:

Chirality (chemistry), a property of molecules having a non-superimposable mirror image

Chirality (electromagnetism), an electromagnetic propagation in chiral media

Chirality (mathematics), the property of a figure not being identical to its mirror image

Chirality (physics), when a phenomenon is not identical to its mirror image

Chirality (journal), an academic journal dealing with chiral chemistry

Chirality (manga), a 4-volume yuri manga series written and illustrated by author Satoshi Urushihara

Chirality (album), a 2014 solo piano album by American pianist John Burke

Chirality (mathematics)

In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. An object that is not chiral is said to be achiral. In 3 dimensions, not all achiral objects have a mirror plane. For example, a 3-dimensional object with inversion centre as its only nontrivial symmetry operation is achiral but has no mirror plane.A chiral object and its mirror image are said to be enantiomorphs. The word chirality is derived from the Greek χείρ (cheir), the hand, the most familiar chiral object; the word enantiomorph stems from the Greek ἐναντίος (enantios) 'opposite' + μορφή (morphe) 'form'. A non-chiral figure is called achiral or amphichiral.


Two-dimensional rotation can occur in two possible directions. A clockwise (typically abbreviated as CW) motion is one that proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back up to the top. The opposite sense of rotation or revolution is (in North American English) counterclockwise (CCW) or (in Commonwealth English) anticlockwise (ACW).


In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. These particles obey the Pauli exclusion principle. Fermions include all quarks and leptons, as well as all composite particles made of an odd number of these, such as all baryons and many atoms and nuclei. Fermions differ from bosons, which obey Bose–Einstein statistics.

A fermion can be an elementary particle, such as the electron, or it can be a composite particle, such as the proton. According to the spin-statistics theorem in any reasonable relativistic quantum field theory, particles with integer spin are bosons, while particles with half-integer spin are fermions.

In addition to the spin characteristic, fermions have another specific property: they possess conserved baryon or lepton quantum numbers. Therefore, what is usually referred to as the spin statistics relation is in fact a spin statistics-quantum number relation.As a consequence of the Pauli exclusion principle, only one fermion can occupy a particular quantum state at any given time. If multiple fermions have the same spatial probability distribution, then at least one property of each fermion, such as its spin, must be different. Fermions are usually associated with matter, whereas bosons are generally force carrier particles, although in the current state of particle physics the distinction between the two concepts is unclear. Weakly interacting fermions can also display bosonic behavior under extreme conditions. At low temperature fermions show superfluidity for uncharged particles and superconductivity for charged particles.

Composite fermions, such as protons and neutrons, are the key building blocks of everyday matter.

The name fermion was coined by English theoretical physicist Paul Dirac from the surname of Italian physicist Enrico Fermi.

Flavour (particle physics)

In particle physics, flavour or flavor refers to the species of an elementary particle. The Standard Model counts six flavours of quarks and six flavours of leptons. They are conventionally parameterized with flavour quantum numbers that are assigned to all subatomic particles. They can also be described by some of the family symmetries proposed for the quark-lepton generations.

Index of physics articles (C)

The index of physics articles is split into multiple pages due to its size.

To navigate by individual letter use the table of contents below.

Relativistic quantum mechanics

In physics, relativistic quantum mechanics (RQM) is any Poincaré covariant formulation of quantum mechanics (QM). This theory is applicable to massive particles propagating at all velocities up to those comparable to the speed of light c, and can accommodate massless particles. The theory has application in high energy physics, particle physics and accelerator physics, as well as atomic physics, chemistry and condensed matter physics. Non-relativistic quantum mechanics refers to the mathematical formulation of quantum mechanics applied in the context of Galilean relativity, more specifically quantizing the equations of classical mechanics by replacing dynamical variables by operators. Relativistic quantum mechanics (RQM) is quantum mechanics applied with special relativity. Although the earlier formulations, like the Schrödinger picture and Heisenberg picture were originally formulated in a non-relativistic background, a few of them (e.g. the Heisenberg formulism) also works with special relativity.

Key features common to all RQMs include: the prediction of antimatter, electron spin, spin magnetic moments of elementary spin 1/2 fermions, fine structure, and quantum dynamics of charged particles in electromagnetic fields. The key result is the Dirac equation, from which these predictions emerge automatically. By contrast, in non-relativistic quantum mechanics, terms have to be introduced artificially into the Hamiltonian operator to achieve agreement with experimental observations.

The most successful (and most widely used) RQM is relativistic quantum field theory (QFT), in which elementary particles are interpreted as field quanta. A unique consequence of QFT that has been tested against other RQMs is the failure of conservation of particle number, for example in matter creation and annihilation.In this article, the equations are written in familiar 3D vector calculus notation and use hats for operators (not necessarily in the literature), and where space and time components can be collected, tensor index notation is shown also (frequently used in the literature), in addition the Einstein summation convention is used. SI units are used here; Gaussian units and natural units are common alternatives. All equations are in the position representation; for the momentum representation the equations have to be Fourier transformed – see position and momentum space.

Spin (physics)

In quantum mechanics and particle physics, spin is an intrinsic form of angular momentum carried by elementary particles, composite particles (hadrons), and atomic nuclei.Spin is one of two types of angular momentum in quantum mechanics, the other being orbital angular momentum. The orbital angular momentum operator is the quantum-mechanical counterpart to the classical angular momentum of orbital revolution and appears when there is periodic structure to its wavefunction as the angle varies. The existence of spin angular momentum is inferred from experiments, such as the Stern–Gerlach experiment, in which silver atoms were observed to possess two possible discrete angular momenta despite having no orbital angular momentum.In some ways, spin is like a vector quantity; it has a definite magnitude, and it has a "direction" (but quantization makes this "direction" different from the direction of an ordinary vector). All elementary particles of a given kind have the same magnitude of spin angular momentum, which is indicated by assigning the particle a spin quantum number.The SI unit of spin is the (N·m·s) or (kg·m2·s−1), just as with classical angular momentum. In practice, spin is given as a dimensionless spin quantum number by dividing the spin angular momentum by the reduced Planck constant ħ, which has the same units of angular momentum, although this is not the full computation of this value. Very often, the "spin quantum number" is simply called "spin", leaving its meaning as the unitless "spin quantum number" to be inferred from context.

When combined with the spin-statistics theorem, the spin of electrons results in the Pauli exclusion principle, which in turn underlies the periodic table of chemical elements.

Wolfgang Pauli in 1924 was the first to propose a doubling of electron states due to a two-valued non-classical "hidden rotation". In 1925, George Uhlenbeck and Samuel Goudsmit at Leiden University suggested the simple physical interpretation of a particle spinning around its own axis, in the spirit of the old quantum theory of Bohr and Sommerfeld. Ralph Kronig anticipated the Uhlenbeck-Goudsmit model in discussion with Hendrik Kramers several months earlier in Copenhagen, but did not publish. The mathematical theory was worked out in depth by Pauli in 1927. When Paul Dirac derived his relativistic quantum mechanics in 1928, electron spin was an essential part of it.

Spock Must Die!

Spock Must Die! is an American science fiction novel written by James Blish, published February 1970 by Bantam Books. It was the first original novel based on the Star Trek television series intended for adult readers. It was preceded by a tie-in comic book line published by Gold Key and the novel Mission to Horatius by Mack Reynolds, all intended for younger readers.Blish aimed to kill off a popular character as a way to surprise readers, and during the novel's production chose Spock, with the aid of his wife, J.A. Lawrence.

Reviews of the novel have been mixed. Some reviewers have directed criticism at the structure or tone of the novel, while others have expressed no enthusiasm for the work, overall.Spock Must Die! was reprinted numerous times with different cover art, including a cover by Kazuhiko Sano. The novel was collected in an omnibus for the Science Fiction Book Club in 1978.

Prior to the release of the Spock Must Die!, Blish had written three collections of short stories adapting episodes of the television series. The second collection, Star Trek 2 (February 1968), included an adaptation of the episode "Errand of Mercy", which the novel directly references in the second chapter.

C, P, and T symmetries

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