Bispinor

A bispinor, also known as a Dirac spinor, is a four-component complex mathematical object in quantum field theory that describes fundamental spin-1/2 fermions, such as electrons, quarks, and other elementary particles, under relativistic conditions.^[1]^[2] It combines two two-component Weyl spinors of opposite chirality—a left-handed and a right-handed spinor—into a single entity that transforms according to the spinor representation of the Lorentz group, ensuring invariance under rotations, boosts, and parity transformations.^[1]^[2] Introduced by Paul Dirac in 1928 as part of his relativistic wave equation, the bispinor provides a framework for incorporating both quantum mechanics and special relativity, resolving issues like the negative probability densities in the Klein-Gordon equation.^[2] The Dirac equation, (iγ^μ ∂_μ - m)ψ = 0, where ψ is the bispinor field and γ^μ are the Dirac matrices, governs its behavior and predicts the existence of antimatter, such as positrons, as distinct particles with positive energy.^[2] In the standard representation, the bispinor ψ can be decomposed as ψ = (φ_R, χ_L)^T, where φ_R and χ_L are the right- and left-chiral components, each with two complex entries, yielding eight real degrees of freedom that correspond to particle and antiparticle states after quantization.^[1] Under Lorentz transformations, bispinors transform via 4×4 matrices S(Λ) derived from the double cover SL(2,ℂ) of the Lorentz group, such that a 2π rotation yields ψ → -ψ, requiring a 4π rotation for the identity—a hallmark of spinorial double-valuedness.^[2]^[1] This structure ensures the bispinor encodes both the four-momentum and four-spin of the particle, with observables like the four-velocity U_μ = ψ† γ^0 γ^μ ψ and four-spin W_μ proportional to the particle's mass and spin properties.^[1] In quantum field theory, bispinor fields are quantized to create fermionic creation and annihilation operators, forming the basis for the Standard Model's description of weak and electromagnetic interactions involving chiral projections.^[2] Despite their abstract nature, bispinors have been experimentally validated through phenomena like electron spin-orbit coupling and the discovery of antimatter in 1932.^[2]

Fundamentals

Definition

A bispinor, also known as a Dirac spinor, is a four-component complex vector in relativistic quantum mechanics that transforms under the reducible representation (1/2, 0) \oplus (0, 1/2) of the Lorentz group SL(2,ℂ).^[3] This representation combines the two fundamental spinor representations of the Lorentz group, allowing the bispinor to encode both left- and right-handed degrees of freedom for spin-1/2 particles.^[4] The bispinor can be understood as the direct sum of a left-handed Weyl spinor, transforming in the (1/2, 0) representation, and a right-handed Weyl spinor, transforming in the (0, 1/2) representation.^[3] In standard notation, it is expressed as

\psi = \begin{pmatrix} \psi_L \\ \psi_R \end{pmatrix},

where \psi_L and \psi_R are two-component complex spinors representing the left- and right-chiral components, respectively.^[4] This structure arises naturally in the chiral basis, where the projectors P_L = (1 - \gamma_5)/2 and P_R = (1 + \gamma_5)/2 isolate \psi_L = P_L \psi and \psi_R = P_R \psi.^[3] In physical contexts, bispinors play a central role in describing massive spin-1/2 fermions, such as electrons and quarks, in four-dimensional Minkowski spacetime, accommodating both chiralities to enable Lorentz-invariant mass terms like m \bar{\psi} \psi.^[3] This formulation is essential for the Dirac equation, which governs the relativistic dynamics of these particles.^[4]

Relation to Spinors and Clifford Algebras

The concept of spinors originated with the need to describe the intrinsic angular momentum, or spin, of particles like the electron in non-relativistic quantum mechanics. In 1927, Wolfgang Pauli introduced two-component spinors, known as Pauli spinors, to represent the spin-1/2 degree of freedom, transforming under the SU(2) double cover of the rotation group SO(3).^[5] These spinors provided a faithful representation of spin operators via the Pauli matrices but were inadequate for relativistic contexts due to the lack of Lorentz invariance.^[5] This construction was motivated by Paul Dirac's 1928 quest for a relativistic wave equation that yielded positive-definite probabilities, avoiding the negative probability densities arising in the second-order Klein-Gordon equation for spin-0 particles.^[6] Dirac's linear first-order equation naturally required four components—the bispinor—to satisfy both the Hamiltonian form and Lorentz covariance, resolving the interpretational issues of the Klein-Gordon theory.^[7] Bispinors combine a left-handed Weyl spinor with a right-handed Weyl spinor into a four-component object, allowing the Dirac mass term to couple these chiralities dynamically.^[8] To address certain issues in the massive case, Hermann Weyl proposed in 1929 two-component spinors, called Weyl spinors, which transform under the (1/2, 0) or (0, 1/2) representations of the Lorentz group SL(2,ℂ). These chiral spinors describe left- or right-handed helicity states for massless particles, where the two components correspond to distinct irreps of the Lorentz algebra.^[5] Algebraically, bispinors find their rigorous foundation in Clifford algebras, specifically the real Clifford algebra Cl(1,3) associated with Minkowski spacetime of signature (1,3), which has dimension 2^{1+3} = 16.^[9] This algebra is generated by spacetime vectors satisfying the anticommutation relations {γ^μ, γ^ν} = 2η^{μν}, and the even subalgebra contains the bivectors that generate the Lorentz algebra, with the spin group—a double cover of the Lorentz group—acting on the bispinors via 4×4 complex matrices in the spinor representation.^[10] Equivalently, the complexified Cl_4(ℂ) underpins the Dirac algebra, ensuring the 16-dimensional structure accommodates both spin and spacetime degrees of freedom.^[9]

Transformations and Properties

Lorentz Transformations

Bispinors transform under the Lorentz group SO(1,3) through its universal cover SL(2,\mathbb{C}), providing a faithful representation of spacetime symmetries in quantum field theory for spin-1/2 particles. The transformation law for a bispinor \psi under a Lorentz transformation \Lambda is given by \psi'(x') = S(\Lambda) \psi(\Lambda^{-1} x'), where x' = \Lambda x and S(\Lambda) is a $4 \times 4 unitary matrix in the bispinor space that preserves the Dirac equation's covariance.^[2] This representation realizes the double cover of the proper orthochronous Lorentz group SO^+(1,3), ensuring that rotations by $4\pi (rather than $2\pi) return the spinor to itself, a hallmark of half-integer spin.^[11] The general form of the transformation operator is S(\Lambda) = \exp\left(-\frac{i}{4} \omega_{\mu\nu} \sigma^{\mu\nu}\right), where \omega_{\mu\nu} are the antisymmetric Lorentz transformation parameters and \sigma^{\mu\nu} = \frac{i}{2} [\gamma^\mu, \gamma^\nu] with \gamma^\mu the Dirac gamma matrices. For infinitesimal transformations, this expands to S(\Lambda) \approx I - \frac{i}{4} \omega_{\mu\nu} \sigma^{\mu\nu}, where I is the identity matrix. The \sigma^{\mu\nu} act as the infinitesimal generators of the Lorentz transformations, spanning the Lie algebra \mathfrak{so}(1,3) in the four-dimensional bispinor space and satisfying the commutation relations [\sigma^{\mu\nu}, \sigma^{\rho\sigma}] = i (\eta^{\nu\rho} \sigma^{\mu\sigma} - \eta^{\mu\rho} \sigma^{\nu\sigma} - \eta^{\nu\sigma} \sigma^{\mu\rho} + \eta^{\mu\sigma} \sigma^{\nu\rho}), with \eta^{\mu\nu} the Minkowski metric.^[12]^[2] Lorentz transformations decompose into rotations and boosts. A spatial rotation by angle \theta around unit axis \mathbf{n} is represented by S(R) = \exp\left(-i \frac{\theta}{2} \mathbf{n} \cdot \boldsymbol{\sigma}\right), where \boldsymbol{\sigma} are the Pauli matrices embedded in the bispinor structure. A pure boost with rapidity \phi along direction \mathbf{n} has the explicit form

S(B) = \cosh\left(\frac{\phi}{2}\right) I - \sinh\left(\frac{\phi}{2}\right) (\boldsymbol{\alpha} \cdot \mathbf{n}),

where \boldsymbol{\alpha}^i = \gamma^0 \gamma^i are the Dirac alpha matrices; this hyperbolic structure arises from exponentiating the boost generators and ensures covariance for massive particles.^[2]^[13] Under the proper orthochronous Lorentz group, bispinors admit a chiral decomposition into left- and right-handed components, \psi = \begin{pmatrix} \psi_L \\ \psi_R \end{pmatrix}, where \psi_{L/R} = \frac{1 \mp \gamma^5}{2} \psi with \gamma^5 = i \gamma^0 \gamma^1 \gamma^2 \gamma^3. The left-handed part \psi_L transforms under the fundamental (1/2, 0) representation of SL(2,\mathbb{C}), while the right-handed \psi_R transforms under the conjugate (0, 1/2) representation, decoupling the transformations for massless limits but mixing under massive conditions. This structure underscores the bispinor's role as the direct sum of these two irreducible representations of the Lorentz algebra.^[2]^[14]

Algebraic Properties

A bispinor, or Dirac spinor, is an element of the four-dimensional complex vector space \mathbb{C}^4, providing a representation space for the (1/2, 0) ⊕ (0, 1/2) irreducible representation of the Lorentz group SL(2, \mathbb{C}). This structure allows bispinors to encode both left- and right-handed chiral components, essential for describing relativistic fermions. The inner product on this space is defined by the Dirac conjugate, \bar{\psi} = \psi^\dagger \gamma^0, yielding the invariant scalar \bar{\psi} \psi = \psi^\dagger \gamma^0 \psi, which is Hermitian and preserves the positive-definite norm for physical states. This bilinear form ensures unitarity in quantum mechanical interpretations and facilitates the construction of observables. Bispinors give rise to Lorentz-covariant bilinears, which transform as tensors under the Lorentz group. The scalar bilinear \bar{\psi} \psi is invariant, the pseudoscalar \bar{\psi} i \gamma^5 \psi changes sign under parity, the vector current \bar{\psi} \gamma^\mu \psi transforms as a four-vector, the axial vector \bar{\psi} \gamma^\mu \gamma^5 \psi as an axial four-vector, the antisymmetric tensor \bar{\psi} \sigma^{\mu\nu} \psi (with \sigma^{\mu\nu} = \frac{i}{2} [\gamma^\mu, \gamma^\nu]) as a rank-2 tensor, and the pseudoscalar counterpart \bar{\psi} i \sigma^{\mu\nu} \gamma^5 \psi accordingly. These bilinears satisfy Fierz identities, such as relations among their squares and products, reflecting the algebraic closure of the Clifford algebra underlying the \gamma-matrices. Under parity transformation, the bispinor transforms as P \psi(t, \mathbf{x}) P^{-1} = \gamma^0 \psi(t, -\mathbf{x}), mapping the upper and lower components while inverting spatial coordinates to preserve the Dirac equation's form. For charge conjugation, relevant to Majorana-like conditions where the particle is its own antiparticle, the operation is C \psi = i \gamma^2 \psi^*, which exchanges particle and antiparticle components and satisfies C^\dagger = -C with C \gamma^\mu C^{-1} = -(\gamma^\mu)^T. The algebraic structure includes a completeness relation for the basis states, where the sum over the four orthonormal bispinors \{\psi_i\} (spanning \mathbb{C}^4) resolves the identity via \sum_i \psi_i \bar{\psi}_i = I, ensuring a complete basis; in the context of field expansions, this extends to momentum-space relations yielding \delta^4(p_i - p_j) for plane-wave modes.

Derivation of Representations

Gamma Matrices

The gamma matrices, denoted \gamma^\mu for \mu = 0, 1, 2, 3, form the core algebraic structure for bispinors in four-dimensional Minkowski spacetime, enabling the representation of Lorentz-invariant fermionic fields.^[2] They are 4×4 complex matrices satisfying the defining anticommutation relations

\{ \gamma^\mu, \gamma^\nu \} = \gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2 g^{\mu\nu} I_4,

where g^{\mu\nu} = \mathrm{diag}(1, -1, -1, -1) is the Minkowski metric tensor in the (+---) signature and I_4 denotes the 4×4 identity matrix.^[2] These relations ensure that (\gamma^0)^2 = I_4 and (\gamma^k)^2 = -I_4 for the spatial indices k = 1, 2, 3, reflecting the signature's distinction between time and space components.^[2] The Hermitian conjugation properties of the gamma matrices are crucial for preserving probability currents in the Dirac equation: (\gamma^0)^\dagger = \gamma^0 for the Hermitian time component, and (\gamma^k)^\dagger = -\gamma^k for the anti-Hermitian spatial components.^[2] These ensure that the combination \gamma^\mu aligns with the requirements of a unitary quantum theory under Lorentz transformations. A key derived object is the chiral projector \gamma^5, defined by

\gamma^5 = i \gamma^0 \gamma^1 \gamma^2 \gamma^3.

It satisfies (\gamma^5)^2 = I_4 and anticommutes with every \gamma^\mu, i.e., \{ \gamma^5, \gamma^\mu \} = 0, allowing the decomposition of bispinors into left- and right-handed chiral components via the projectors P_L = \frac{1 - \gamma^5}{2} and P_R = \frac{1 + \gamma^5}{2}.^[2] In the Hermitian sense, (\gamma^5)^\dagger = \gamma^5.^[2] The explicit construction in the standard Weyl (or chiral) basis highlights the bispinor structure as a direct sum of two-component Weyl spinors:

\gamma^0 = \begin{pmatrix} 0 & I_2 \\ I_2 & 0 \end{pmatrix}, \quad \gamma^k = \begin{pmatrix} 0 & \sigma^k \\ -\sigma^k & 0 \end{pmatrix},

where I_2 is the 2×2 identity and \sigma^k (k=1,2,3) are the Pauli matrices \sigma^1 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \sigma^2 = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \sigma^3 = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.^[2] In this basis, \gamma^5 = \begin{pmatrix} -I_2 & 0 \\ 0 & I_2 \end{pmatrix}, which is diagonal and separates the chiral sectors.^[2] These matrices generate the Clifford algebra Cl(1,3) associated with the Lorentz group, providing the algebraic foundation for bispinor transformations.^[15]

Embedding of Lorentz Algebra

The embedding of the Lorentz Lie algebra \mathfrak{so}(1,3) into the Clifford algebra \mathrm{Cl}(1,3) is realized through bilinear combinations of the gamma matrices \gamma^\mu, which serve as the fundamental generators of the algebra satisfying \{\gamma^\mu, \gamma^\nu\} = 2 g^{\mu\nu} I. The six generators of \mathfrak{so}(1,3) are given by

J^{\mu\nu} = \frac{i}{4} [\gamma^\mu, \gamma^\nu],

where the commutator ensures antisymmetry in \mu,\nu, and the factor of i aligns with the standard Hermitian form in the spinor representation.^[16] These J^{\mu\nu} act on the bispinor space and faithfully represent the infinitesimal Lorentz transformations. The generators satisfy the defining commutation relations of the Lorentz Lie algebra:

[J^{\mu\nu}, J^{\rho\sigma}] = i \left( g^{\mu\rho} J^{\nu\sigma} + g^{\nu\sigma} J^{\mu\rho} - g^{\mu\sigma} J^{\nu\rho} - g^{\nu\rho} J^{\mu\sigma} \right),

which follow directly from the anticommutation properties of the \gamma^\mu and confirm the embedding within the Clifford structure. This closure under commutation demonstrates that the Lorentz algebra is a subalgebra of the even-grade elements in \mathrm{Cl}(1,3). The rotation subgroup \mathfrak{so}(3) \subset \mathfrak{so}(1,3), corresponding to spatial indices i,j=1,2,3, is generated by J^{ij}. These can be recast in vector form as J^{ij} = \frac{1}{2} \epsilon^{ijk} \Sigma^k, where the spin operators are \Sigma^k = \frac{i}{2} [\gamma^l, \gamma^m] with \{l,m,k\} a cyclic permutation of \{1,2,3\}, establishing the isomorphism \mathfrak{so}(3) \cong \mathfrak{su}(2) in the bispinor representation.^[16] The boost generators, which mix time and space, are the mixed components K^i = J^{0i} = \frac{i}{2} \gamma^0 \gamma^i, arising from the anticommutation \{\gamma^0, \gamma^i\}=0 that distinguishes the Lorentz metric. This representation is faithful and irreducible on the 4-dimensional complex bispinor space, reflecting the isomorphism \mathfrak{so}(1,3) \cong \mathfrak{sl}(2,\mathbb{C}), with the generators embedded as $4 \times 4 complex matrices from the complexification \mathrm{Cl}(1,3;\mathbb{R}) \otimes \mathbb{C} \cong \mathrm{Cl}_4(\mathbb{C}) \cong M_4(\mathbb{C}).

Construction of Bispinor Basis

The bispinor space, also known as the Dirac spinor space, is a four-dimensional complex vector space equipped with a basis constructed from solutions to the Dirac equation for free particles of definite momentum and helicity. In the Dirac representation of the gamma matrices, the basis vectors for positive-energy solutions are given by

u(\mathbf{p}, s) = \sqrt{E + m} \begin{pmatrix} \chi^s \\ \frac{\vec{\sigma} \cdot \mathbf{p}}{E + m} \chi^s \end{pmatrix},

where E = \sqrt{\mathbf{p}^2 + m^2} is the energy, m is the particle mass, \vec{\sigma} are the Pauli matrices, and \chi^s (s = 1, 2) are normalized two-component spinors satisfying \chi^{s\dagger} \chi^{s'} = \delta_{ss'}, such as \chi^1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix} and \chi^2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix} for spin along the z-axis.^[17] For negative-energy solutions, corresponding to antiparticles, the basis vectors are

v(\mathbf{p}, s) = \sqrt{E + m} \begin{pmatrix} \frac{\vec{\sigma} \cdot \mathbf{p}}{E + m} \chi^s \\ -\chi^s \end{pmatrix}.

These forms ensure that the spinors satisfy the Dirac equation (\slash{p} - m) u(\mathbf{p}, s) = 0 for positive energy and (\slash{p} + m) v(\mathbf{p}, s) = 0 for negative energy, with the overall normalization factor \sqrt{E + m} chosen to achieve the relativistic normalization u^\dagger(\mathbf{p}, s) u(\mathbf{p}, s') = 2E \delta_{ss'}.^[17] The covariant normalization, relevant for Lorentz-invariant quantities, is \bar{u}(\mathbf{p}, s) u(\mathbf{p}, s') = 2m \delta_{ss'}, where \bar{u} = u^\dagger \gamma^0, and similarly \bar{v}(\mathbf{p}, s) v(\mathbf{p}, s') = -2m \delta_{ss'}. This normalization arises from the structure of the spinors and the properties of the Dirac matrices in the representation where \gamma^0 is diagonal. The positive- and negative-energy projectors are defined as \Lambda^+(\mathbf{p}) = \frac{\slash{p} + m}{2m} and \Lambda^-(\mathbf{p}) = \frac{m - \slash{p}}{2m}, which satisfy \Lambda^+(\mathbf{p}) u(\mathbf{p}, s) = u(\mathbf{p}, s) and \Lambda^-(\mathbf{p}) v(\mathbf{p}, s) = v(\mathbf{p}, s), projecting onto the respective eigenspaces of the Dirac operator.^[17] In the chiral (Weyl) basis, the bispinor \psi decomposes as \psi = \psi_L + \psi_R, where \psi_{L/R} are the left- and right-handed chiral components projected by P_{L/R} = \frac{1 \mp \gamma^5}{2}, with \gamma^5 = i \gamma^0 \gamma^1 \gamma^2 \gamma^3. This decomposition separates the upper and lower two-components in the chiral representation, facilitating analysis of parity-violating interactions, though the basis construction remains equivalent to the Dirac representation up to a unitary transformation.^[2] The set \{u(\mathbf{p}, s), v(\mathbf{p}, s)\} for s = 1, 2 forms a complete basis for the bispinor space at fixed on-shell momentum p^2 = m^2, satisfying the completeness relation \sum_s \left[ u(\mathbf{p}, s) \bar{u}(\mathbf{p}, s) - v(\mathbf{p}, s) \bar{v}(\mathbf{p}, s) \right] = 2m I_4. This relation, derived from the projectors, ensures that any bispinor can be expanded in this basis, with the difference reflecting the distinction between particle and antiparticle contributions.^[17]

Dirac Algebra Integration

Conventions and Dirac Matrices

In the formulation of bispinors, which are four-component spinors satisfying the Dirac equation, the choice of representation for the Dirac matrices \gamma^\mu plays a crucial role in simplifying calculations and highlighting physical properties such as chirality or reality conditions.^[18] The major conventions include the Dirac (standard), Weyl (chiral), and Majorana (real) representations, each providing a basis for the 4×4 \gamma^\mu matrices that satisfy the Clifford algebra \{\gamma^\mu, \gamma^\nu\} = 2\eta^{\mu\nu} I_4, where \eta^{\mu\nu} is the Minkowski metric.^[19] These representations are related by similarity transformations and are chosen based on the context, such as emphasizing the distinction between left- and right-handed components in the Weyl basis or enabling real-valued spinors in the Majorana basis.^[20] The Dirac representation, also known as the standard or Dirac-Pauli representation, is widely used in non-relativistic limits and quantum electrodynamics due to its block-diagonal structure for \gamma^0. In this convention, with the metric signature (+---),

\gamma^0 = \begin{pmatrix} I_2 & 0 \\ 0 & -I_2 \end{pmatrix}, \quad \gamma^k = \begin{pmatrix} 0 & \sigma^k \\ -\sigma^k & 0 \end{pmatrix},

where I_2 is the 2×2 identity matrix and \sigma^k (for k=1,2,3) are the Pauli matrices.^[19] This form ensures \gamma^0 is Hermitian and \gamma^k are anti-Hermitian, preserving the hermiticity properties essential for a unitary theory.^[18] In the Weyl representation, the matrices are off-diagonal, facilitating the separation into chiral components:

\gamma^0 = \begin{pmatrix} 0 & I_2 \\ I_2 & 0 \end{pmatrix}, \quad \gamma^k = \begin{pmatrix} 0 & \sigma^k \\ -\sigma^k & 0 \end{pmatrix},

which highlights the massless limit where left- and right-handed bispinors decouple.^[19] The Majorana representation employs purely real (or imaginary) matrices, allowing the bispinor to be self-conjugate, \psi = \psi^c, useful for theories with Majorana fermions like neutralinos.^[18] The choice of metric signature, predominantly (+---) in particle physics versus (-+++) in some general relativity contexts, influences the signs in the Dirac equation and the hermiticity of the \gamma^\mu. With (+---), the Dirac equation is i \gamma^\mu \partial_\mu \psi - m \psi = 0, where \partial_\mu = (\partial_t, -\nabla) and the mass term remains positive, ensuring a stable vacuum.^[18] Switching to (-+++) requires adjusting signs, such as \gamma^0 \to i \gamma^0 in some formulations, to maintain probability conservation and positive energy solutions, though this can complicate interactions with electromagnetic fields.^[21] These metric choices affect bispinor bilinears, like the scalar \bar{\psi} \psi, by altering overall signs but preserve Lorentz invariance.^[22] Different representations are interconvertible via similarity transformations S \gamma^\mu S^{-1} = \gamma'^\mu, where S is a 4×4 invertible matrix ensuring the Clifford algebra is preserved. For instance, the transformation from Dirac to Weyl basis involves S = \sqrt{i \gamma^0 \gamma^5} (up to phases), allowing seamless switching without altering physical predictions.^[20] Such equivalences underscore that bispinor properties, like parity transformation \psi \to \gamma^0 \psi, remain representation-independent.^[19]

Spinor Construction Examples

In the Dirac representation, a concrete example of bispinor construction is the positive-energy solution for an electron at rest with spin up along the z-axis. This spinor takes the form

u(0, \uparrow) = \sqrt{2m} \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix},

where m is the electron mass, ensuring normalization \bar{u} u = 2m.^[23] This form arises from solving the Dirac equation at zero momentum, where the upper two components correspond to the large Pauli spinor for spin up, and the lower components vanish.^[24] To construct a bispinor for a boosted electron, the rest-frame spinor is transformed using the Lorentz boost operator D(\Lambda), which for a boost along the z-direction with momentum \mathbf{p} = (0, 0, p_z) and energy E = \sqrt{m^2 + p_z^2} yields

u(p, \uparrow) = \sqrt{E + m} \begin{pmatrix} 1 \\ 0 \\ \frac{p_z}{E + m} \\ 0 \end{pmatrix}.

This explicit application preserves the spin orientation in the rest frame while accounting for relativistic effects, such as the mixing between upper and lower components.^[24] The boost operator is D(p) = \sqrt{\frac{E + m}{2m}} \begin{pmatrix} I_2 & \frac{\mathbf{p} \cdot \sigma}{E + m} \\ \frac{\mathbf{p} \cdot \sigma}{E + m} & I_2 \end{pmatrix}, applied to the rest-frame form.^[23] Charge conjugation provides another construction method, transforming an electron bispinor into a positron state via the operator C = i \gamma^2 \gamma^0. For the rest-frame electron spinor above, the conjugated positron spinor is v(0, \uparrow) = C \bar{u}(0, \uparrow)^T = \sqrt{2m} \begin{pmatrix} 0 \\ 0 \\ 0 \\ -1 \end{pmatrix} (up to phase conventions), effectively flipping the charge while reversing the energy sign from positive to negative in the Dirac sea interpretation.^[25] This operation satisfies C \gamma^\mu C^{-1} = -(\gamma^\mu)^T, ensuring the Dirac equation invariance under charge reversal.^[23] Helicity eigenstates for bispinors, particularly in the ultra-relativistic or massless limit where helicity aligns with chirality, are constructed using the projectors involving \gamma^5. The state with helicity h = +1/2 (right-handed) is \psi_{+1/2} = \frac{1 + \gamma^5}{2} \psi, and for h = -1/2 (left-handed), \psi_{-1/2} = \frac{1 - \gamma^5}{2} \psi, where \psi is a general Dirac spinor and \gamma^5 = i \gamma^0 \gamma^1 \gamma^2 \gamma^3.^[2] These projectors yield Weyl spinors as eigenstates of the helicity operator \frac{1}{2} \hat{\mathbf{p}} \cdot \boldsymbol{\Sigma} with eigenvalues \pm 1/2.^[23]