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physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, and specifically in
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
, a bispinor, is a mathematical construction that is used to describe some of the
fundamental particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be elementary include electrons, the fundamental fermions (quarks, leptons, antiqu ...
s of
nature Nature, in the broadest sense, is the physics, physical world or universe. "Nature" can refer to the phenomenon, phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. ...
, including
quark A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly o ...
s and
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no kn ...
s. It is a specific embodiment of a
spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight ...
, specifically constructed so that it is consistent with the requirements of
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The laws o ...
. Bispinors transform in a certain "spinorial" fashion under the action of the
Lorentz group In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
, which describes the symmetries of
Minkowski spacetime In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inert ...
. They occur in the relativistic spin-
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
solutions to the
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac part ...
. Bispinors are so called because they are constructed out of two simpler component spinors, the
Weyl spinor In physics, particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three p ...
s. Each of the two component spinors transform differently under the two distinct complex-conjugate spin-1/2
representations ''Representations'' is an interdisciplinary journal in the humanities published quarterly by the University of California Press. The journal was established in 1983 and is the founding publication of the New Historicism movement of the 1980s. It ...
of the Lorentz group. This pairing is of fundamental importance, as it allows the represented particle to have a
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
, carry a
charge Charge or charged may refer to: Arts, entertainment, and media Films * '' Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * ''Charge!!'', an album by The Aqu ...
, and represent the flow of charge as a
current Currents, Current or The Current may refer to: Science and technology * Current (fluid), the flow of a liquid or a gas ** Air current, a flow of air ** Ocean current, a current in the ocean *** Rip current, a kind of water current ** Current (stre ...
, and perhaps most importantly, to carry
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
. More precisely, the mass is a
Casimir invariant In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operator ...
of the Lorentz group (an eigenstate of the energy), while the vector combination carries momentum and current, being covariant under the action of the Lorentz group. The angular momentum is carried by the
Poynting vector In physics, the Poynting vector (or Umov–Poynting vector) represents the directional energy flux (the energy transfer per unit area per unit time) or '' power flow'' of an electromagnetic field. The SI unit of the Poynting vector is the watt ...
, suitably constructed for the spin field. A bispinor is more or less "the same thing" as a
Dirac spinor In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos. It appears in the plane-wave solution to the Dirac equation, and is a certain combin ...
. The convention used here is that the article on the Dirac spinor presents
plane-wave In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space. For any position \vec x in space and any time t, t ...
solutions to the Dirac equation using the Dirac convention for the
gamma matrices In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\ma ...
. That is, the Dirac spinor is a bispinor in the Dirac convention. By contrast, the article below concentrates primarily on the Weyl, or chiral representation, is less focused on the Dirac equation, and more focused on the geometric structure, including the geometry of the
Lorentz group In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
. Thus, much of what is said below can be applied to the
Majorana equation In physics, the Majorana equation is a relativistic wave equation. It is named after the Italian physicist Ettore Majorana, who proposed it in 1937 as a means of describing fermions that are their own antiparticle. Particles corresponding to this ...
.


Definition

Bispinors are elements of a 4-dimensional
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
representation of the
Lorentz group In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
. In the
Weyl basis In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\ma ...
, a bispinor :\psi = \left(\begin\psi_\\ \psi_\end\right) consists of two (two-component) Weyl spinors \psi_ and \psi_ which transform, correspondingly, under (, 0) and (0, ) representations of the \mathrm(1,3) group (the Lorentz group without
parity transformation In physics, a parity transformation (also called parity inversion) is the flip in the sign of ''one'' spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (a point refle ...
s). Under parity transformation the Weyl spinors transform into each other. The Dirac bispinor is connected with the Weyl bispinor by a unitary transformation to the
Dirac basis In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\ma ...
, : \psi \rightarrow \left begin 1 & 1 \\ -1 & 1 \end\rightpsi = \left(\begin \psi_ + \psi_ \\ \psi_ - \psi_ \end\right) . The Dirac basis is the one most widely used in the literature.


Expressions for Lorentz transformations of bispinors

A bispinor field \psi(x) transforms according to the rule :\psi^a(x) \to ^a\left(x^\prime\right) = S
Lambda Lambda (}, ''lám(b)da'') is the 11th letter of the Greek alphabet, representing the voiced alveolar lateral approximant . In the system of Greek numerals, lambda has a value of 30. Lambda is derived from the Phoenician Lamed . Lambda gave rise ...
a_b \psi^b\left(\Lambda^x^\prime\right) = S
Lambda Lambda (}, ''lám(b)da'') is the 11th letter of the Greek alphabet, representing the voiced alveolar lateral approximant . In the system of Greek numerals, lambda has a value of 30. Lambda is derived from the Phoenician Lamed . Lambda gave rise ...
a_b \psi^b(x) where \Lambda is a
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
. Here the coordinates of physical points are transformed according to x^\prime = \Lambda x, while S, a matrix, is an element of the spinor representation (for spin ) of the Lorentz group. In the Weyl basis, explicit transformation matrices for a boost \Lambda_ and for a rotation \Lambda_ are the following:David Tong
''Lectures on Quantum Field Theory''
(2012), Lecture 4
:\begin S Lambda_&= \left(\begin e^ & 0 \\ 0 & e^ \end\right) \\ S Lambda_&= \left(\begin e^ & 0 \\ 0 & e^ \end\right) \end Here \chi is the boost parameter, and \phi^i represents rotation around the x^i axis. \sigma_i are the
Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in ...
. The exponential is the exponential map, in this case the
matrix exponential In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives ...
defined by putting the matrix into the usual power series for the exponential function.


Properties

A
bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called ''scalars''). In other words, a bilinear form is a function that is linear i ...
of bispinors can be reduced to five irreducible (under the Lorentz group) objects: #
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
, \bar\psi; #
pseudo-scalar In linear algebra, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion while a true scalar does not. Any scalar product between a pseudovector and an ordinary vector is a pseudoscalar. Th ...
, \bar\gamma^5\psi; #
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
, \bar\gamma^\mu\psi; # pseudo-vector, \bar\gamma^\mu\gamma^5\psi; #
antisymmetric tensor In mathematics and theoretical physics, a tensor is antisymmetric on (or with respect to) an index subset if it alternates sign (+/−) when any two indices of the subset are interchanged. section §7. The index subset must generally either be all ' ...
, \bar\left(\gamma^\mu\gamma^\nu - \gamma^\nu\gamma^\mu\right)\psi, where \bar \equiv \psi^\dagger\gamma^0 and \left\ are the
gamma matrices In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\ma ...
. These five quantities are inter-related by the Fierz identities. Their values are used in the Lounesto spinor field classification of the different types of spinors, of which the bispinor is just one; the others being the
flagpole A flagpole, flagmast, flagstaff, or staff is a pole designed to support a flag. If it is taller than can be easily reached to raise the flag, a cord is used, looping around a pulley at the top of the pole with the ends tied at the bottom. The fla ...
(of which the
Majorana spinor In physics, the Majorana equation is a relativistic wave equation. It is named after the Italian physicist Ettore Majorana, who proposed it in 1937 as a means of describing fermions that are their own antiparticle. Particles corresponding to this e ...
is a special case), the flag-dipole, and the
Weyl spinor In physics, particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three p ...
. The flagpole, flag-dipole and Weyl spinors all have null mass and pseudoscalar fields; the flagpole additionally has a null pseudovector field, whereas the Weyl spinors have a null antisymmetric tensor (a null "angular momentum field"). A suitable
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
for the relativistic spin- field can be built out of these, and is given as :\mathcal = \left(\bar\gamma^\mu\partial_\mu\psi - \partial_\mu\bar\gamma^\mu\psi\right) - m\bar\psi\;. The
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac part ...
can be derived from this Lagrangian by using the
Euler–Lagrange equation In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...
.


Derivation of a bispinor representation


Introduction

This outline describes one type of bispinors as elements of a particular representation space of the (, 0) ⊕ (0, ) representation of the Lorentz group. This representation space is related to, but not identical to, the representation space contained in the
Clifford algebra In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperc ...
over
Minkowski spacetime In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inert ...
as described in the article
Spinors In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight ...
. Language and terminology is used as in
Representation theory of the Lorentz group 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 representati ...
. The only property of Clifford algebras that is essential for the presentation is the defining property given in below. The basis elements of are labeled . A representation of the Lie algebra of the Lorentz group will emerge among matrices that will be chosen as a basis (as a vector space) of the complex Clifford algebra over spacetime. These matrices are then exponentiated yielding a representation of . This representation, that turns out to be a representation, will act on an arbitrary 4-dimensional complex vector space, which will simply be taken as , and its elements will be bispinors. For reference, the commutation relations of are with the spacetime metric .


The gamma matrices

Let denote a set of four 4-dimensional gamma matrices, here called the ''Dirac matrices''. The Dirac matrices satisfy where is the
anticommutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, a ...
, is a unit matrix, and is the spacetime metric with signature (+,−,−,−). This is the defining condition for a generating set of a
Clifford algebra In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperc ...
. Further basis elements of the Clifford algebra are given by Only six of the matrices are linearly independent. This follows directly from their definition since . They act on the subspace the span in the passive sense, according to In , the second equality follows from property of the Clifford algebra.


Lie algebra embedding of so(3,1) in Cl4(C)

Now define an action of on the , and the linear subspace ''they'' span in , given by The last equality in , which follows from and the property of the gamma matrices, shows that the constitute a representation of since the
commutation relation In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
s in are exactly those of . The action of can either be thought of as six-dimensional matrices multiplying the basis vectors , since the space in spanned by the is six-dimensional, or be thought of as the action by commutation on the . In the following, The and the are both (disjoint) subsets of the basis elements of Cl4(C), generated by the four-dimensional Dirac matrices in four spacetime dimensions. The Lie algebra of is thus embedded in Cl4(C) by as the ''real'' subspace of Cl4(C) spanned by the . For a full description of the remaining basis elements other than and of the Clifford algebra, please see the article
Dirac algebra In mathematical physics, the Dirac algebra is the Clifford algebra \text_(\mathbb). This was introduced by the mathematical physicist P. A. M. Dirac in 1928 in developing the Dirac equation for spin-½ particles with a matrix representation of t ...
.


Bispinors introduced

Now introduce ''any'' 4-dimensional complex vector space ''U'' where the ''γ''''μ'' act by matrix multiplication. Here will do nicely. Let be a Lorentz transformation and ''define'' the action of the Lorentz group on ''U'' to be :u \rightarrow S(\Lambda)u = e^u;\quad u^\alpha \rightarrow ^\alpha_\beta u^\beta. Since the according to constitute a representation of , the induced map according to general theory either is a representation or a
projective representation In the field of representation theory in mathematics, a projective representation of a group ''G'' on a vector space ''V'' over a field ''F'' is a group homomorphism from ''G'' to the projective linear group \mathrm(V) = \mathrm(V) / F^*, where GL(' ...
of . It will turn out to be a projective representation. The elements of ''U'', when endowed with the transformation rule given by ''S'', are called bispinors or simply spinors.


A choice of Dirac matrices

It remains to choose a set of Dirac matrices in order to obtain the spin representation . One such choice, appropriate for the
ultrarelativistic limit In physics, a particle is called ultrarelativistic when its speed is very close to the speed of light . The expression for the relativistic energy of a particle with rest mass and momentum is given by :E^2 = m^2 c^4 + p^2 c^2. The energy of a ...
, is where the are the
Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in ...
. In this representation of the Clifford algebra generators, the become This representation is manifestly ''not'' irreducible, since the matrices are all
block diagonal In mathematics, a block matrix or a partitioned matrix is a matrix that is '' interpreted'' as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original m ...
. But by irreducibility of the Pauli matrices, the representation cannot be further reduced. Since it is a 4-dimensional, the only possibility is that it is a representation, i.e. a bispinor representation. Now using the recipe of exponentiation of the Lie algebra representation to obtain a representation of , a projective 2-valued representation is obtained. Here is a vector of rotation parameters with , and is a vector of boost parameters. With the conventions used here one may write for a bispinor field. Here, the upper component corresponds to a ''right''
Weyl spinor In physics, particularly in quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions. The equation is named after Hermann Weyl. The Weyl fermions are one of the three p ...
. To include space parity inversion in this formalism, one sets as representative for . It is seen that the representation is irreducible when space parity inversion is included.


An example

Let so that generates a rotation around the ''z''-axis by an angle of . Then but . Here, denotes the identity element. If is chosen instead, then still , but now . This illustrates the double-valued nature of a spin representation. The identity in gets mapped into either or depending on the choice of Lie algebra element to represent it. In the first case, one can speculate that a rotation of an angle negates a bispinor, and that it requires a rotation to rotate a bispinor back into itself. What really happens is that the identity in is mapped to ''in'' with an unfortunate choice of . It is impossible to continuously choose for all so that is a continuous representation. Suppose that one defines along a loop in such that . This is a closed loop in , i.e. rotations ranging from 0 to around the ''z''-axis under the exponential mapping, but it is only "half" a loop in , ending at . In addition, the value of is ambiguous, since and gives different values for .


The Dirac algebra

The representation on bispinors will induce a representation of on , the set of linear operators on ''U''. This space corresponds to the Clifford algebra itself so that all linear operators on ''U'' are elements of the latter. This representation, and how it decomposes as a direct sum of irreducible representations, is described in the article on
Dirac algebra In mathematical physics, the Dirac algebra is the Clifford algebra \text_(\mathbb). This was introduced by the mathematical physicist P. A. M. Dirac in 1928 in developing the Dirac equation for spin-½ particles with a matrix representation of t ...
. One of the consequences is the decomposition of the bilinear forms on . This decomposition hints how to couple any bispinor field with other fields in a Lagrangian to yield
Lorentz scalar In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of ...
s.


Bispinors and the Dirac algebra

The
Dirac matrices In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\ma ...
are a set of four 4×4
matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
forming the
Dirac algebra In mathematical physics, the Dirac algebra is the Clifford algebra \text_(\mathbb). This was introduced by the mathematical physicist P. A. M. Dirac in 1928 in developing the Dirac equation for spin-½ particles with a matrix representation of t ...
, and are used to intertwine the
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...
direction with the
local reference frame In theoretical physics, a local reference frame (local frame) refers to a coordinate system or frame of reference that is only expected to function over a small region or a restricted region of space or spacetime. The term is most often used in ...
(the local coordinate frame of spacetime), as well as to define
charge Charge or charged may refer to: Arts, entertainment, and media Films * '' Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * ''Charge!!'', an album by The Aqu ...
(
C-symmetry In physics, charge conjugation is a transformation that switches all particles with their corresponding antiparticles, thus changing the sign of all charges: not only electric charge but also the charges relevant to other forces. The term C-symm ...
), parity and time reversal
operators Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another sp ...
.


Conventions

There are several choices of
signature A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
and representation that are in common use in the physics literature. The Dirac matrices are typically written as \gamma^\mu where \mu runs from 0 to 3. In this notation, 0 corresponds to time, and 1 through 3 correspond to ''x'', ''y'', and ''z''. The
signature A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
is sometimes called the
west coast West Coast or west coast may refer to: Geography Australia * Western Australia *Regions of South Australia#Weather forecasting, West Coast of South Australia * West Coast, Tasmania **West Coast Range, mountain range in the region Canada * Britis ...
metric, while the is the
east coast East Coast may refer to: Entertainment * East Coast hip hop, a subgenre of hip hop * East Coast (ASAP Ferg song), "East Coast" (ASAP Ferg song), 2017 * East Coast (Saves the Day song), "East Coast" (Saves the Day song), 2004 * East Coast FM, a ra ...
metric. At this time the signature is in more common use, and our example will use this signature. To switch from one example to the other, multiply all \gamma^\mu by i. After choosing the signature, there are many ways of constructing a representation in the 4×4 matrices, and many are in common use. In order to make this example as general as possible we will not specify a representation until the final step. At that time we will substitute in the "chiral" or Weyl representation.


Construction of Dirac spinor with a given spin direction and charge

First we choose a
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...
direction for our electron or positron. As with the example of the Pauli algebra discussed above, the spin direction is defined by a
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vecto ...
in 3 dimensions, (a, b, c). Following the convention of Peskin & Schroeder, the spin operator for spin in the (a, b, c) direction is defined as the dot product of (a, b, c) with the vector :\begin \left(i\gamma^2\gamma^3,\;\;i\gamma^3\gamma^1,\;\;i\gamma^1\gamma^2\right) &= -\left(\gamma^1,\;\gamma^2,\;\gamma^3\right)i\gamma^1\gamma^2\gamma^3 \\ \sigma_ &= ia\gamma^2\gamma^3 + ib\gamma^3\gamma^1 + ic\gamma^1\gamma^2 \end Note that the above is a
root of unity In mathematics, a root of unity, occasionally called a Abraham de Moivre, de Moivre number, is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematic ...
, that is, it squares to 1. Consequently, we can make a
projection operator In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...
from it that projects out the sub-algebra of the Dirac algebra that has spin oriented in the (a, b, c) direction: :P_ = \frac\left(1 + \sigma_\right) Now we must choose a charge, +1 (positron) or −1 (electron). Following the conventions of Peskin & Schroeder, the operator for charge is Q = -\gamma^0, that is, electron states will take an eigenvalue of −1 with respect to this operator while positron states will take an eigenvalue of +1. Note that Q is also a square root of unity. Furthermore, Q commutes with \sigma_. They form a
complete set of commuting operators In quantum mechanics, a complete set of commuting observables (CSCO) is a set of commuting operators whose common eigenvectors can be used as a basis to express any quantum state. In the case of operators with discrete spectra, a CSCO is a set of c ...
for the
Dirac algebra In mathematical physics, the Dirac algebra is the Clifford algebra \text_(\mathbb). This was introduced by the mathematical physicist P. A. M. Dirac in 1928 in developing the Dirac equation for spin-½ particles with a matrix representation of t ...
. Continuing with our example, we look for a representation of an electron with spin in the direction. Turning Q into a projection operator for charge = −1, we have :P_ = \frac\left(1 - Q\right) = \frac\left(1 + \gamma^0\right) The projection operator for the spinor we seek is therefore the product of the two projection operators we've found: :P_\;P_ The above projection operator, when applied to any spinor, will give that part of the spinor that corresponds to the electron state we seek. So we can apply it to a spinor with the value 1 in one of its components, and 0 in the others, which gives a column of the matrix. Continuing the example, we put (''a'', ''b'', ''c'') = (0, 0, 1) and have :P_ = \frac\left(1 + i\gamma_1\gamma_2\right) and so our desired projection operator is :P = \frac\left(1+ i\gamma^1\gamma^2\right) \cdot \frac\left(1 + \gamma^0\right) = \frac\left(1 + \gamma^0 + i\gamma^1\gamma^2 + i\gamma^0\gamma^1\gamma^2\right) The 4×4 gamma matrices used in the Weyl representation are :\begin \gamma_0 &= \begin0 & 1 \\ 1 & 0\end \\ \gamma_k &= \begin0 & \sigma^k \\ -\sigma^k & 0\end \end for ''k'' = 1, 2, 3 and where \sigma^i are the usual 2×2
Pauli matrices In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in ...
. Substituting these in for ''P'' gives :P = \frac\begin1 + \sigma^3 & 1 + \sigma^3 \\ 1 + \sigma^3 & 1 + \sigma^3 \end = \frac\begin1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0\end Our answer is any non-zero column of the above matrix. The division by two is just a normalization. The first and third columns give the same result: :\left, e^-,\, +\frac\right\rangle = \begin 1 \\ 0 \\ 1 \\ 0 \end More generally, for electrons and positrons with spin oriented in the (''a'', ''b'', ''c'') direction, the projection operator is :\frac\begin 1 + c & a - ib & \pm(1 + c) & \pm(a - ib) \\ a + ib & 1 - c & \pm(a + ib) & \pm(1 - c) \\ \pm(1 + c) & \pm(a - ib) & 1 + c & a - ib \\ \pm(a + ib) & \pm(1 - c) & a + ib & 1 - c \end where the upper signs are for the electron and the lower signs are for the positron. The corresponding spinor can be taken as any non zero column. Since a^2 + b^2 + c^2 = 1 the different columns are multiples of the same spinor. The representation of the resulting spinor in the
Dirac basis In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\ma ...
can be obtained using the rule given in the bispinor article.


See also

*
Dirac spinor In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos. It appears in the plane-wave solution to the Dirac equation, and is a certain combin ...
* Spin(3,1), the double cover of SO(3,1) by a
spin group In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1. As a L ...
*
Rarita–Schwinger equation In theoretical physics, the Rarita–Schwinger equation is the relativistic field equation of spin-3/2 fermions. It is similar to the Dirac equation for spin-1/2 fermions. This equation was first introduced by William Rarita and Julian Schwinge ...


Notes


References

* *{{citation, last=Weinberg, first=S, year=2002, title=The Quantum Theory of Fields, vol I, isbn=0-521-55001-7, url-access=registration, url=https://archive.org/details/quantumtheoryoff00stev. Quantum field theory Spinors