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An active transformation keeps the coordinate argument fixed and changes a contravariant vector field by $A^{\prime \mu }(x)={\Lambda }^{\mu }{}_{\nu }{A}^{\nu }({\Lambda }^{-1}x)$. Infinitesimally, $\delta A^{\mu }=\alpha {\omega }^{\mu }{}_{\nu }{A}^{\nu }-\alpha {\omega }^{\rho }{}_{\sigma }{x}^{\sigma }{\partial }_{\rho }{A}^{\mu }$. For a Lorentz scalar density, the spin-index term is absent, giving $\delta \mathcal{L}=-\alpha {\omega }^{\rho }{}_{\sigma }{x}^{\sigma }{\partial }_{\rho }\mathcal{L}$.

Since the chirality matrix commutes with spinor Lorentz generators, $j_5^{\mu }=\bar{\psi }{\gamma }^{\mu }{\gamma }^5\psi$ transforms as a vector under proper Lorentz transformations and as a pseudovector under parity. Contracting two copies gives the Lorentz scalar $j_{5\mu }{j}_5^{\mu }$; the extra parity signs cancel. Lorentz invariance of this interaction does not require current conservation.

A central charge in four-dimensional extended supersymmetry is a Lorentz scalar even generator appearing in $\{Q_{\alpha }^A,Q_{\beta }^B\}={ϵ}_{\alpha \beta }{Z}^{AB}$. It commutes with the displayed translation, Lorentz algebra, and supercharge generators. An added R-symmetry automorphism can act on the central charges.

The generators in the Spinor representation of the Lorentz group satisfy $M^{†}{\gamma }^0+{\gamma }^{0}M=0$, so the finite transformation preserves the indefinite form $S^{†}{\gamma }^{0}S={\gamma }^0$. This makes $\bar{\psi }\psi$ a Lorentz scalar through the Dirac adjoint. Boost generators are Hermitian rather than anti-Hermitian: a boost has eigenvalues $e^{\pm \eta /2}$ and is not unitary in the ordinary positive-definite component norm. Rotations are unitary. This component representation is distinct from the unitary action on the physical state space.

The four-divergence of a four-vector field is the Lorentz scalar contraction ${\partial }_{\mu }{V}^{\mu }$. With $x^{\mu }=(ct,\mathbf{x})$, it equals $c^{-1}{\partial }_t{V}^0+\nabla \cdot \mathbf{V}$. The four-current has zero four-divergence, expressing local charge conservation.

Under CPT, a Hermitian local tensor monomial picks up a minus sign for each Lorentz vector index, including derivative indices. A Lorentz scalar contracts all such indices, so the total sign is positive; charged terms are exchanged with their Hermitian conjugates. Therefore a local Hermitian Lagrangian density obeys $\Theta \mathcal{L}(x){\Theta }^{-1}=\mathcal{L}(-x)$, leaving its action invariant. This field-level argument illustrates the CPT theorem for ordinary scalar, spinor and vector theories; the axiomatic theorem also uses positive energy and microcausality.

The source-free electromagnetic Lagrangian density is quadratic in the electromagnetic field tensor. Its Euler-Lagrange field equation is ${\partial }_{\mu }{F}^{\mu \nu }=0$. Antisymmetry of $F$ makes the density invariant under gauge transformations, and contraction of its indices makes it a Lorentz scalar.

Lowering indices with the stated metric reverses the sign of each time-space component and leaves each space-space component unchanged. Contracting the two antisymmetric tensors, counting each unordered index pair twice, givesThese are independent Lorentz scalars; their signs follow from the explicit dual convention in the PDF.

Put $J=E\cdot B>0$ and $D=|E{|}^2-|B{|}^2$. In the parallel-field frame, invariance gives $E^{\prime }{B}^{\prime }=J$ and $E^{\prime 2}-B^{\prime 2}=D$. The positive product forces the two vectors to have the same orientation, so the use of positive magnitudes is legitimate. Substitution of $B^{\prime }=J/E^{\prime }$ gives $E^{\prime 4}-D{E}^{\prime 2}-J^2=0$. Its positive solution is $E^{\prime 2}=(D+\sqrt{D^2+4J^2})/2$. With $K=D/(2J)$,The denominator is positive for every real $K$. The other quadratic root is negative and cannot be a squared magnitude. No explicit boost transformation is needed because the two invariants determine the magnitudes uniquely in a frame where the fields are parallel.

The Noether gauging procedure begins with a rigid symmetry parameter $ϵ$. Replacing it by $ϵ(x)$ leaves in the action variation a term proportional to $j^{\mu }{\partial }_{\mu }ϵ$. Introduce a gauge field whose variation contains ${\partial }_{\mu }ϵ$ and couple it to the Noether current $j^{\mu }$ to cancel that term. The new coupling produces further variations, so both the action and transformation laws are corrected successively in the coupling constant until local invariance is obtained. Covariant derivatives organize this iterative completion.

For the Wess–Zumino model, the parameter is a spinor and the current is the supercurrent. Its derivative part has the structure ${\gamma }^{\nu }{\partial }_{\nu }(A-i{\gamma }_{5}B){\gamma }^{\mu }\psi$, up to the fixed Majorana spinor normalization and improvements. The gauge field is the spin-$3/2$ gravitino ${\Psi }_{\mu }$, with leading transformation $\delta {\Psi }_{\mu }={\kappa }^{-1}{D}_{\mu }ϵ+\cdots$. A coupling of ${\bar{\Psi }}_{\mu }$ to that current cancels the derivative of the local parameter. Since the commutator of two local supersymmetries includes a spacetime transformation, the vierbein and gravity must participate as well. This gives the Wess-Zumino chiral multiplet coupled to supergravity.

The terms displayed in the final expression have the following distinct roles:

The last two interactions are part of the higher-order local supersymmetry completion. They can be organized through the composite Kähler connection in the fermion derivatives, or derived by eliminating the corresponding supergravity auxiliary connection. Matter fermions and gravitini carry the appropriate chiral Kähler weights, fixing the relative current couplings. They are not new adjustable Yukawa interactions. In reading this organization, one must not include the composite connection in $D_{\mu }$ and then count its separately displayed current terms a second time.

Finally the spin connection itself acquires gravitino-induced torsion and matter spin-current contributions. Eliminating this algebraic connection, or solving for auxiliary fields, can make further fermion contact terms implicit in the chosen covariant derivatives and component convention. The key mechanism is current coupling at first order, followed by the covariant and higher-order terms required by local supersymmetry, rather than merely changing ordinary derivatives in the original matter Lagrangian.

For $x^{\prime }=\Lambda x$, choose the corresponding spin liftThe spinor field transforms asThe infinitesimal Clifford algebra identity from part (b) exponentiates to $S^{-1}{\gamma }^{\mu }S={\Lambda }^{\mu }{}_{\nu }{\gamma }^{\nu }$, establishing Lorentz covariance of the Dirac operator. At fixed coordinates the infinitesimal variation also includes the orbital term $-{\Omega }^{\mu }{}_{\nu }{x}^{\nu }{\partial }_{\mu }\psi$.

A spatial rotation generator $M^{ij}$ is anti-Hermitian, but a Lorentz boost generator is Hermitian. For example,Its eigenvalues are $\pm \frac{1}{2}$. A nonzero rapidity $\zeta$ gives boost eigenvalues $e^{\pm \zeta /2}$, which are not all on the unit circle. This proves that the finite-dimensional component Spinor representation of the Lorentz group cannot be unitary for a positive-definite component inner product, even after a change of basis. It does not contradict a unitary action on the physical space of quantum states.

Consequently ${\psi }^{\prime †}(x^{\prime }){\psi }^{\prime }(x^{\prime })={\psi }^{†}{S}^{†}S\psi$ need not equal ${\psi }^{†}\psi$: it is the time component of the Dirac current, not a Lorentz scalar. The appropriate invariant form is indefinite. From $({\gamma }^{\mu }{)}^{†}={\gamma }^0{\gamma }^{\mu }{\gamma }^0$ one obtainsThis Dirac spinor pseudo-unitarity gives ${\bar{\psi }}^{\prime }(x^{\prime })=\bar{\psi }(x)S^{-1}$ and hence the Dirac scalar bilinear

The CPT theorem is a structural consequence of relativistic quantum field theory. In flat spacetime, assume a positive physical state space, the positive-energy spectrum condition, Lorentz invariance, microcausality, the usual Spin-statistics theorem, and a Hermitian dynamics. For ordinary field theories these are realized by a local Hermitian Lorentz-scalar Lagrangian density. There is then an antiunitary operator $\Theta$ implementing the combined charge conjugation, parity symmetry in quantum field theory and time-reversal symmetry, with a CPT-invariant vacuum. The combined symmetry is required even when the three separate symmetries fail.

Under the combined spacetime transformation, $x\mapsto -x$. A complex scalar field can be taken to transform into its adjoint at $-x$; a Hermitian vector field transforms into minus itself at $-x$. A derivative contributes a minus sign through differentiation of its reflected argument. For a Dirac field the transformation supplied in the paper may be writtenField phases are conventional and do not affect the physical conclusions. Crucially, antiunitarity means $\Theta i{\Theta }^{-1}=-i$ and conjugates every numerical coefficient. The reversal of time is compatible with positive energies because this conjugation changes $e^{-iHt}$ to $e^{iHt}$ while leaving the Hamiltonian invariant. Treating time reversal as a unitary change of coordinates would instead give incorrect signs.

One can see explicitly how the CPT transformation of Dirac bilinears works. Let $B_{\Gamma }(x)=:{\psi }^{†}{\gamma }^0\Gamma \psi :$ be a Hermitian fermion bilinear, with ${\Gamma }^{†}={\gamma }^0\Gamma {\gamma }^0$. The colons indicate normal ordering so that the graded reordering below has no vacuum contact term. Using the reflected fields and conjugating ${\gamma }^0\Gamma$, then interchanging the two fermion factors, givesThe last step uses the Dirac-Hermiticity condition and $\{{\gamma }^0,{\gamma }_5\}=0$. The identities for the gamma matrices then giveThe pseudoscalar's sign is particularly instructive: its explicit $i$ must also be conjugated. The minus sign from exchanging fermions is equally essential. These signs depend on the number of free vector indices, rather than on a field's separate parity assignment.

This supplies a local Lagrangian argument for CPT invariance. Hermitian local field combinations with $r$ vector indices transform with sign $(-1{)}^r$, including the appropriate reordered derivative terms. A Lorentz scalar contracts those indices in pairs or with four-index invariant tensors, giving a positive total sign. For charged terms, the reflected conjugate fields and conjugated coefficients exchange a term with its Hermitian-conjugate partner. ThusThe integration variable $x\mapsto -x$ has unit absolute Jacobian. The general field-index argument is discussed in Srednicki's treatment of discrete symmetries, section 40.

For example, the scalar mass term $\overline{\psi }\psi$ is even. The Dirac electromagnetic current and vector potential are both odd, making their contracted interaction even. The Hermitian Dirac kinetic term $(i/2)\overline{\psi }{\gamma }^{\mu }{\overleftrightarrow{\partial }}_{\mu }\psi$ is also even: reflection and reordering interchange the two derivative placements, while antiunitarity conjugates its $i$. A complex Yukawa coupling and its Hermitian conjugate are exchanged rather than requiring the coupling itself to be real. Hence CP violation in the CKM matrix is compatible with exact CPT. Renormalizability is not necessary: local Hermitian higher-dimensional Lorentz-scalar operators satisfy the same argument.

The axiomatic form of the CPT theorem does not start by assuming a polynomial Lagrangian density. The positive-energy spectrum condition provides analytic continuations of Wightman functions, and complexified Lorentz transformations relate configurations to their total spacetime inversions. At suitable spacelike configurations, microcausality permits reversal of the order of field operators with the signs dictated by the Spin-statistics theorem. Uniqueness of analytic continuation extends this relation to the correlators generally, from which the Hilbert-space antiunitary operator is obtained. This explains why locality and positive energy are genuine hypotheses, not incidental features of a particular interaction. The elementary argument above displays the mechanism for the scalar, spinor and vector fields relevant to the Standard Model; it is not a substitute for all the analytic steps of the axiomatic proof.

For one-particle states, CPT reverses additive charges and spin projections. The spatial momentum is unchanged by the combination: parity symmetry in quantum field theory and time reversal each reverse it. Thus, up to phases,Since the energy-momentum spectrum is preserved, the CPT constraints on particle properties give equal particle-antiparticle masses, equal spin magnitudes, and opposite conserved additive charges. Relating the corresponding two-point functions also fixes equal pole positions, so unstable conjugate particles have equal total decay widths, hence equal lifetimes. Magnetic moments at the same chosen spin orientation have opposite signs. These are precise experimental tests of CPT, not tests that every individual discrete symmetry is respected.

For scattering, CPT interchanges incoming and outgoing boundary conditions. The antiunitary operator identity $⟨a|b⟩=⟨\Theta b|\Theta a⟩$ relates $⟨f,\mathrm{out}|i,\mathrm{in}⟩$ to the amplitude with transformed $f$ incoming and transformed $i$ outgoing. Thus the reversed process with antiparticles and reversed spins has the same transition probability. This is not, in general, equality with the forward CP-conjugate process. In particular, CP violation can make corresponding partial decay widths different while their complete sums remain equal. With CPT preserved, genuine CP violation requires corresponding time-reversal symmetry violation.

Finally, a proposed CPT violation requires revisiting at least one assumption: for example, a nonlocal interaction, a preferred spacetime direction violating Lorentz invariance, or loss of unitary Hermitian dynamics. A generic curved background need not possess the global spacetime inversion used here. Those observations delimit the theorem; they do not weaken its prediction for the ordinary local Standard Model in Minkowski spacetime.

Use left Grassmann derivatives and define the supersymmetric covariant derivatives $D_{\pm }={\partial }_{{\theta }^{\pm }}-i{\theta }^{\pm }{\partial }_{\pm }$. Multiplication in the indicated order gives exactly the printed differential operator:Thus the equation is $i{D}_{-}{D}_{+}\Phi =W(\Phi )$ in two-dimensional N=(1,1) superspace.

Under a Lorentz boost, take $x^{\pm }\mapsto e^{\pm \omega }{x}^{\pm }$ and ${\theta }^{\pm }\mapsto e^{\pm \omega /2}{\theta }^{\pm }$. The derivatives ${\partial }_{\pm }$ have boost weights $\mp 1$, and $D_{\pm }$ have weights $\mp 1/2$. Consequently $D_{-}{D}_{+}$ has weight zero, and is a Lorentz scalar. Both $\Phi$ and the ordinary function $W(\Phi )$ are scalar superfields, so the equation is Lorentz invariant.

For supersymmetry, $\{{\partial }_{\theta },\theta \}=1$ and anticommutation of distinct odd coordinates giveMoving either supercharge through the two odd derivatives introduces two minus signs, hence $[Q_t,D_{-}{D}_{+}]=0$. If $\delta \Phi ={ϵ}^{+}{Q}_{+}\Phi +{ϵ}^{-}{Q}_{-}\Phi$, with odd constant parameters, the chain rule for an even scalar superfield givesTherefore a solution is mapped to a solution, proving supersymmetry invariance.

To compute the supersymmetric Liouville equation, order the odd coordinates as ${\theta }^{-}{\theta }^{+}$ and anticommute the fermionic component fields with both coordinates. In particular, ${\partial }_{{\theta }^{+}}({\theta }^{-}{\theta }^{+})=-{\theta }^{-}$. Direct application of the operator givesLet $N=\Phi -\varphi$. Only the two products of its degree-one terms survive in $N^2$; for example, $(i{\theta }^{-}{\psi }_{+})(i{\theta }^{+}{\psi }_{-})={\theta }^{-}{\theta }^{+}{\psi }_{+}{\psi }_{-}$. Hence $N^2=2{\theta }^{-}{\theta }^{+}{\psi }_{+}{\psi }_{-}$ and $N^3=0$. The exponential function expands exactly asEquating the constant terms fixes the auxiliary field, $\boxed{F=e^{\varphi }}$. The linear and quadratic terms then give the requested component field equations of motion:The signs here follow from the printed operator, the printed component expansion, and left Grassmann derivatives; no convention change has been made in eliminating the auxiliary field.

Here L-superfield means a left-chiral superfield: ${\bar{D}}_{\dot{\alpha }}\Phi =0$, and its complex conjugate is an antichiral superfield, $D_{\alpha }{\Phi }^{\ast }=0$. Interpret the two barred derivatives as the contracted scalar ${\bar{D}}^2={\bar{D}}_{\dot{\alpha }}{\bar{D}}^{\dot{\alpha }}$. The key point is chiral projection by squared supercovariant derivatives. Since the two components of $\bar{D}$ anticommute, any product of three barred derivatives repeats a component and is zero. Thusfor every superfield $X$, even when $X$ is antichiral. In particular, ${\bar{D}}^2{\Phi }^{\ast }$ is chiral.

The right side is chiral too. Because $\Phi$ is even, the graded derivation rule gives ${\bar{D}}_{\dot{\alpha }}({\Phi }^2)=({\bar{D}}_{\dot{\alpha }}\Phi )\Phi +\Phi ({\bar{D}}_{\dot{\alpha }}\Phi )=0$. Constant $m$ and $\lambda$ do not contribute derivatives. Consequentlyso applying the chiral constraint to the equation imposes no additional incompatible condition. This is the holomorphic closure of chiral superfields; a generic nonholomorphic right side would not satisfy it.

Both sides have zero Grassmann parity and are Lorentz scalars: the two dotted derivative indices are contracted with the invariant epsilon tensor, and $\Phi$ is a scalar superfield. Moreover, each covariant derivative anticommutes with every supercharge, so its squared contraction commutes with the supercharges. Hence the equation transforms covariantly under supersymmetry. It is the chiral field equation of motion of the Wess–Zumino model with superpotential $\mathcal{W}(\Phi )=m{\Phi }^2/2+\lambda {\Phi }^3/3$, for which ${\mathcal{W}}^{\prime }=m\Phi +\lambda {\Phi }^2$, using the normalization in the PDF. With mass dimensions $[\Phi ]=1$, $[\bar{D}]=1/2$, $[m]=1$ and $[\lambda ]=0$, each side has dimension two. No component field equations of motion are needed for this consistency argument.

The supplied matrices are the chiral gamma-matrix representation. The Pauli matrix multiplication law implies $\{{\sigma }^i,{\sigma }^j\}=2{\delta }^{ij}{I}_2$. Block multiplication gives $({\gamma }^0{)}^2=I_4$, $\{{\gamma }^0,{\gamma }^i\}=0$, and $\{{\gamma }^i,{\gamma }^j\}=-2{\delta }^{ij}{I}_4$. Hence their Clifford algebra relation isSince partial derivatives commute, multiplying the Dirac equation by $i{\gamma }^{\nu }{\partial }_{\nu }+m$ givesThus every component of the Dirac spinor satisfies $\boxed{(\square +m^2)\psi =0}$, the Klein-Gordon equation with the same mass.

For a Lorentz transformation $x^{\prime }=\Lambda x$, set ${\psi }^{\prime }(x^{\prime })=S(\Lambda )\psi (x)$. The derivative transforms as ${\partial }_{\mu }^{\prime }=({\Lambda }^{-1}{)}^{\nu }{}_{\mu }{\partial }_{\nu }$. With the given spinor-matrix identity,This proves the Lorentz covariance of the Dirac operator explicitly.

For parity, let $x_P=(x^0,-\mathbf{x})$ and ${\psi }_P(x)={\gamma }^0\psi (x_P)$. Spatial differentiation brings a minus sign, which is canceled by ${\gamma }^i{\gamma }^0=-{\gamma }^0{\gamma }^i$. ThereforeSince ${\gamma }^0$ is Hermitian and squares to one, the Dirac adjoint transforms asThe chirality matrix ${\gamma }^5=i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3$ anticommutes with ${\gamma }^0$. The parity transformation of a Dirac bilinear consequently givesThe first bilinear is a Lorentz scalar even under parity, and the second is a pseudoscalar.

In this representation, ${\gamma }^5=\mathrm{diag}(-I_2,I_2)$. Its square is one, so the chiral projectors $P_{\pm }=(1\pm {\gamma }^5)/2$ are idempotent, mutually orthogonal and sum to the identity. They select ${\psi }_L=P_{-}\psi$ and ${\psi }_R=P_{+}\psi$, the two Weyl spinors. Because ${\gamma }^{\mu }{P}_{\pm }=P_{\mp }{\gamma }^{\mu }$, a solution of the massless Dirac equation has each chiral component solving that equation separately. In two-component form,The two chirality sectors are thus independent under massless propagation and proper Lorentz transformations. A nonzero Dirac mass term couples them, while parity exchanges them. This explains why the projectors are particularly useful in the massless theory.

Continue with metric signature $+---$. The Pauli matrices obey $\{{\sigma }^i,{\sigma }^j\}=2{\delta }^{ij}{I}_2$. Block multiplication in the chiral gamma-matrix representation givesandHence $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2g^{\mu \nu }{I}_4$, the required Clifford algebra. Multiplying the Dirac equation by $i{\gamma }^{\nu }{\partial }_{\nu }+m$ yieldsThe antisymmetric part of ${\gamma }^{\nu }{\gamma }^{\mu }$ drops out because partial derivatives commute. Therefore every component of the Dirac spinor satisfies the Klein-Gordon equation. The converse is not true: four arbitrary scalar solutions need not obey the first-order Dirac equation.

The Lorentz group consists of real invertible matrices $\Lambda$ preserving the Minkowski metric: ${\Lambda }^{T}g\Lambda =g$. Writing $\Lambda =I+\omega +O({\omega }^2)$ gives ${\omega }_{\mu \nu }=-{\omega }_{\nu \mu }$, so there are six continuous parameters: three spatial rotation angles and three boost rapidities. The proper orthochronous Lorentz group is the connected component with determinant $+1$ that preserves time orientation. Parity and time reversal are additional discrete operations and are not encoded by the six real parameters in an exponential near the identity.

For the real-generator convention used here, vector generators can be writtenTake ${\Omega }_{\rho \sigma }=-{\Omega }_{\sigma \rho }$ and use the same parameters in the spinor representation. The Lorentz-spinor generators from a Clifford algebra areTo verify the Lorentz algebra, first commute a generator with one gamma matrix:For example, this follows by moving ${\gamma }^{\tau }$ through each product using the Clifford anticommutator. Applying the commutator derivation rule to $\frac{1}{4}[{\gamma }^{\tau },{\gamma }^{\nu }]$ then givesNo extra factor of $i$ belongs in these generators with this commutator convention. Hermitian-generator conventions shift the factors of $i$ into the brackets and the exponential instead.

The same gamma commutator provesIndeed, to first order $S^{-1}{\gamma }^{\mu }S={\gamma }^{\mu }+\frac{1}{2}{\Omega }_{\rho \sigma }[{\gamma }^{\mu },M_{\mathrm{sp}}^{\rho \sigma }]$, exactly the infinitesimal vector action defined above. Exponentiating proves the finite relation. The spinor field transforms as ${\psi }^{\prime }(x^{\prime })=S(\Lambda )\psi (x)$ with $x^{\prime }=\Lambda x$; combining this relation with ${\partial }_{\mu }^{\prime }=({\Lambda }^{-1}{)}^{\nu }{}_{\mu }{\partial }_{\nu }$ preserves the Dirac equation. A $2\pi$ rotation gives $S=-I_4$, so the spinor matrices give the double-cover spin representation, rather than a single-valued ordinary representation of the Lorentz group itself.

For a boost in direction $i$, $M_{\mathrm{sp}}^{0i}=\frac{1}{2}{\gamma }^0{\gamma }^i$ is Hermitian, with eigenvalues $\pm \frac{1}{2}$. A real rapidity $\eta$ therefore produces eigenvalues $e^{\pm \eta /2}$ in $S$, whose moduli are not one. The finite-component spinor representation is not unitary for boosts, although the rotation matrices are unitary. The Lorentz group is noncompact; this does not forbid the unitary infinite-dimensional action on the physical Hilbert space of states.

Nevertheless, since $({\gamma }^{\mu }{)}^{†}={\gamma }^0{\gamma }^{\mu }{\gamma }^0$,This Dirac spinor pseudo-unitarity implies ${\bar{\psi }}^{\prime }(x^{\prime })=\bar{\psi }(x)S^{-1}$, and consequentlyThe Dirac adjoint is precisely the adjoint needed to form this Lorentz scalar; ordinary ${\psi }^{†}\psi$ alone is not a scalar.

Let ${\gamma }^5=i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3=\mathrm{diag}(-I_2,I_2)$. It anticommutes with every gamma matrix and commutes with every $M_{\mathrm{sp}}^{\rho \sigma }$, so $S^{-1}{\gamma }^{5}S={\gamma }^5$. Thus the axial current transforms as a four-vector under proper Lorentz transformations:Under parity it is an axial vector, acquiring the additional pseudovector sign. Contracting two axial currents cancels that sign and uses the invariant metric. Therefore $j_{5\mu }{j}_5^{\mu }$ is compatible with Lorentz invariance, including parity invariance of this contraction. Lorentz invariance of the interaction does not require the axial current to be conserved.

Use signature $\eta =\mathrm{diag}(1,-1,-1,-1)$, the numerical matrices ${\sigma }_{\mu }=(I,{\sigma }_i)$ and ${\bar{\sigma }}_{\mu }=(I,-{\sigma }_i)$ given in the question, and raise genuine tensor indices with $\eta$. A real Minkowski spacetime vector corresponds to the Hermitian matrixFor SL(2,C) matrices, $X^{\prime }=AX{A}^{†}$ is Hermitian and has the same determinant. It therefore induces a real linear Lorentz transformation; composition of the congruence actions agrees with matrix multiplication. The kernel consists of $\pm I$: preservation of $X=I$ makes a kernel element unitary, and preservation of every Hermitian $X$ makes it scalar.

Polar decomposition of an invertible complex matrix connects every determinant-one complex matrix to its unitary factor, so the group is connected. The action gives the Lorentz spinor double cover of the proper orthochronous Lorentz group. It covers rotations via $A_R=e^{-i\vartheta \mathbf{n}\cdot \mathbf{\sigma }/2}$ and boosts via the positive Hermitian matrices below. Every proper orthochronous transformation is a boost followed by a rotation, since one can first match its image of the future unit time vector and then use its rotation stabilizer. The congruence construction does not cover spatial parity or time reversal. In particular parity has determinant $-1$ as a four-vector transformation; all transformations continuously produced from SL(2,C) have determinant $+1$.

Set $c_{\theta }=\cosh (\theta /2)$, $s_{\theta }=\sinh (\theta /2)$ and $N=\mathbf{n}\cdot \mathbf{\sigma }$. The Pauli matrix multiplication law gives $N^2=I$. The proposed boost matrix is $A_B=c_{\theta }I+s_{\theta }N$, with eigenvalues $e^{\pm \theta /2}$, determinant one and $A_B^{†}=A_B$. Split $\mathbf{x}=x_{\parallel }\mathbf{n}+{\mathbf{x}}_{\perp }$. The perpendicular Pauli part anticommutes with $N$, so multiplication givesThis is an active Lorentz boost. The image of a rest worldline has velocity $\mathbf{v}=\tanh \theta \mathbf{n}$, fixing the velocity sign convention.

For two nonzero boosts, write $A^{\prime }=c^{\prime }I+s^{\prime }{\mathbf{n}}^{\prime }\cdot \mathbf{\sigma }$ and $A=cI+s\mathbf{n}\cdot \mathbf{\sigma }$. Their product isThe last term is anti-Hermitian. Thus $A^{\prime }A$ is not Hermitian unless the axes are collinear; a pure boost has Hermitian lifts $\pm A_B$, so it cannot give the same Lorentz transformation. The noncollinear boost obstruction from Pauli products is thereforeThat rotation is a Wigner rotation. A zero-rapidity factor is the trivial exception, regardless of the arbitrary axis assigned to it.

Define $J_i=\frac{1}{2}{ϵ}_{ijk}{M}_{jk}$ and $K_i=M_{0i}$. The printed Lorentz algebra brackets giveSet $L_i=(J_i+i{K}_i)/2$, $R_i=(J_i-i{K}_i)/2$. ThenThis is the chiral decomposition of the complex Lorentz algebra. The two copies are the complexifications of the SU(2) algebras conventionally labelled left and right. They are not two independent compact real subalgebras of the real Lorentz algebra: $L,R$ are complex linear combinations of the real generators. The distinction is needed for noncompact boosts.

To verify the infinitesimal two-component action, putIts trace vanishes by antisymmetry, so $\det A=1+O({\omega }^2)$. The Pauli matrix multiplication law impliesSince $a^{†}=\frac{1}{4}{\omega }^{\mu \nu }{\bar{\sigma }}_{\nu }{\sigma }_{\mu }$, antisymmetrizing this identity yieldsConsequently $AX{A}^{†}=X+{\sigma }_{\mu }{\omega }^{\mu }{}_{\nu }{x}^{\nu }+O({\omega }^2)$, exactly the claimed infinitesimal coordinate transformation with ${\omega }^{\mu }{}_{\nu }={\omega }^{\mu \rho }{\eta }_{\rho \nu }$.

The two Weyl spinor representations transform asBoth maps preserve group multiplication. For $\Lambda =e^{\omega }$, choose its spin lift and write the finite matrices asFor rotations these coincide as the SU(2) doublet; for boosts their generator signs are opposite. Any complex-linear intertwiner would commute with all rotations and hence be scalar by the Schur lemma, but a nonzero scalar cannot intertwine the opposite boost matrices. They are therefore inequivalent. Infinitesimally their $K_i$ matrices are $-i{\sigma }_i/2$ and $+i{\sigma }_i/2$, while their $J_i$ matrices are both ${\sigma }_i/2$. Thus they have chiral labels $(1/2,0)$ and $(0,1/2)$. The finite matrices are representations of the spin cover; choosing $A$ or $-A$ matters for spinors even though their four-vector transformations coincide.

Using the supplied block gamma matrices, the generator of the Dirac spinor isThe antisymmetry of $\omega$ removes the symmetric Clifford part. Hence

There is a conjugation-order error in the last displayed gamma identity in the PDF. The Clifford algebra gives, with ${\gamma }^{\rho }={\eta }^{\rho \sigma }{\gamma }_{\sigma }$,Exponentiating this linear commutator action proves the inverse Lorentz action on gamma matricesThe second identity is the order required for the requested bilinears when $\psi \mapsto S\psi$. An explicit countercheck to the printed order is a positive boost along the third axis: ${\omega }^{03}=-\theta$ giveswhereas the printed right side has a plus sign. This cannot be repaired by dropping index raising; the same raising convention is needed in the preceding coordinate transformation.

The adjoint relation supplied in the question gives ${\Omega }^{†}{\gamma }^0=-{\gamma }^0\Omega$, hence the Dirac spinor pseudo-unitarity identity $S^{†}{\gamma }^{0}S={\gamma }^0$. The Dirac adjoint therefore transforms as $\bar{\psi }\mapsto \bar{\psi }{S}^{-1}$. It now follows that the Dirac scalar bilinear isthe vector isand the antisymmetric second-rank tensor isThese are respectively a Lorentz scalar, Lorentz four-vector and Lorentz tensor. The source's inconsistent gamma identity is replaced by its correct inverse/order pair; all three transformation laws then follow with the stated spinor transformation.

The strong-interaction matrix element between two spin-zero pseudoscalar mesons has only $p^{\mu }$ and $k^{\mu }$ available. The product of the two intrinsic parities is positive. An axial current would require a pseudovector constructed from these momenta, but an expression involving the Levi-Civita symbol needs three independent four-vectors and therefore vanishes. This is a consequence of parity conservation in the hadronic matrix element, not of parity conservation in the weak interaction. The vector current can have the two independent structures $p+k$ and $p-k$. Its coefficients are Lorentz scalars; with $p^2=m_K^2$ and $k^2=m_{\pi }^2$ fixed, their only varying invariant is $s=q^2$. HenceThese are the pseudoscalar-to-pseudoscalar form factors. With relativistically normalized states they are dimensionless. The vanishing axial matrix element and this decomposition explain the two equalities separately.