C-P-T Fractionalization

Discrete spacetime symmetries of parity P or reflection R, and time-reversal T, act naively as $\mathbb{Z}_2$-involutions in the passive transformation on the spacetime coordinates; but together with a charge conjugation C, the total C-P-R-T symmetries have enriched active transformations on fields in representations of the spacetime-internal symmetry groups of quantum field theories (QFTs). In this work, we derive that these symmetries can be further fractionalized, especially in the presence of the fermion parity $(-1)^{\rm F}$. We elaborate on examples including relativistic Lorentz invariant QFTs (e.g., spin-1/2 Dirac or Majorana spinor fermion theories) and nonrelativistic quantum many-body systems (involving Majorana zero modes), and comment on applications to spin-1 Maxwell electromagnetism (QED) or interacting Yang-Mills (QCD) gauge theories. We discover various C-P-R-T-$(-1)^{\rm F}$ group structures, e.g., Dirac spinor is in a projective representation of $\mathbb{Z}_2^{\rm C}\times \mathbb{Z}_2^{\rm P} \times \mathbb{Z}_2^{\rm T}$ but in an (anti)linear representation of an order-16 nonabelian finite group, as the central product between an order-8 dihedral (generated by C and P) or quaternion group and an order-4 group generated by T with T$^2=(-1)^{\rm F}$. The general theme may be coined as C-P-T or C-R-T fractionalization.

T(t, x 1 , . . ., x d )T −1 = x T ≡ (−t, x 1 , . . ., x d ), P(t, x 1 , . . ., x d )P −1 = x P ≡ (t, −x 1 , − . . ., −x d ), (1) R(t, x 1 , . . ., x d )R −1 = x R ≡ (t, −x 1 , + . . ., +x d ), where T flips the time coordinate, P flips all x, but R flips only on one coordinate (here say x 1 ) with respect to a mirror plane (normal to x 1 ).We label the spacetime coordinate component x µ with µ = 0, 1, . . ., d for (d + 1)-spacetime dimensions (denoted as d + 1d).The transformed coordinates are labeled as x , or x µ for each component, with the subscript T/P/R/etc.to indicate which coordinates are transformed.In odddimensional spacetime, the P is in fact a subgroup of a continuous spatial rotational symmetry special orthogonal SO(d) ⊂ SO + (d, 1), thus unluckily P is not an independent discrete symmetry.We should replace P by the reflection R. For example, the CPT theorem [1][2][3][4][5][6] should be called the CRT theorem [7,8] in any general dimension of Minkowski spacetime.In this work, we mainly focus on the even-dimensional spacetime, so we can choose either P or R symmetry.We shall mainly use P to match the major literature, but we will comment about R when necessary.
Charge conjugation C, however, cannot manifest itself under a passive transformation on the spacetime coordinates, but can reveal itself under an active transformation on a particle or field, such as a complex-valued spin-0 Lorentz scalar φ(x) = φ(t, x) (which is a function of the spacetime coordinates).The C colloquially flips between particle and anti-particle sectors, or more generally between energetic excitations and anti-excitations C(excitations)C −1 = (anti-excitations) (2) involving the complex conjugation (denoted * ).The active transformation acts on this Lorentz scalar φ as Tφ(t, x)T −1 = φ T (t, x) = φ(−t, x) = φ(x T ).
All the above transformations, regardless passive or active, naively seem to be only Z 2 -involutions in mathematics, such that twice transformations are the null (do nothing) transformations. 1 Thus it reveals a finite group of order 2 structure, namely Z 2 .In this scalar field example, the C-P-T symmetry form a direct product group Z C 2 × Z P 2 × Z T 2 .One may mistakenly conclude C 2 = P 2 = R 2 = T 2 = +1 and assume they are all commute in general.The essence of our work is to point out that all these "discrete C, P, R, or T symmetries" (which we denote altogether as "C-P-R-T" in short) can form a rich nonabelian finite group structure, in the physical realistic systems pertinent to experiments or theories.We can possibly fractionalize the C-P-R-T group structures further, for the statevectors in quantum mechanics or the fields in classical or quantum field theories (QFTs), in various representations (rep) of the spacetime or internal symmetry groups (denoted as G spacetime and G internal ).
The symmetry fractionalization means the following: the matter field is not in the linear representation of the original symmetry group G, but in the projective representation of G and in the linear representation of the extended total group G.A typical case is illustrated by a group extension 1 → N → G → G → 1 where G is the quotient group while the N is the normal subgroup of the total group G, so G/N = G.A famous example is the gapped 1+1d isospin-1 Haldane chain with G = SO(3) symmetry [9], whose 0+1d boundary can host a two-fold degenerated isospin-1/2 doublet of G = SU (2), with N = Z 2 .Thus this doublet is in a projective rep of G = SO (3), also in a linear rep G = SU (2).
In this work, we will find the analogous C-P-T symmetry fractionalization.For example, in contrast to a spin-0 scalar field's G φ ≡ Z C 2 × Z P 2 × Z T 2 , we uncover an order-16 nonabelian Gψ ≡ for a 3+1d spin-1/2 Dirac field (see the later eq.( 6) for explanations).Remarkably, the fermion parity Z F 2 generated by (−1) F : ψ → −ψ plays a crucial role in the group extension structure 1 → Z F 2 → Gψ → G φ → 1.Thus fermionic systems reveal Z F 2 -enriched structures richer than bosonic systems.This means that Dirac fermion ψ is in a projective rep of G φ , also in an (anti)linear rep of Gψ .(It is antilinear because Gψ contains the antilinear time-reversal symmetry.) This beyond-Z 2 group structure for C-P-R-T is mostly secretly hidden in the literature, and still not yet widely appreciated.(However, a well-known exception is the time-reversal symmetry can be Z TF 4 ⊃ Z F 2 that T 2 = (−1) F in contrast with the usual Z T 2 with T 2 = +1, both have applications to the classification of topological superconductors and insulators, see for instance [8,[12][13][14][15][16][17][18][19][20][21]).Moreover, here we stress various new nonabelian finite group structures for the total C-P-R-T symmetries that have not yet been discovered previously.Below we work through several examples in sections.
3+1D SPIN-1/2 FERMIONIC SPINORS First, we consider the 3+1d Dirac theory with a 4 complex component spinor field ψ.We aim to carry out its C-P-R-T-(−1) F structure acting on ψ in detail.It is convenient to regard the massless Dirac spinor as two complex Weyl spinors 2 L ⊕ 2 R (left L and right R) rep in the standard Weyl basis for ψ [22][23][24][25].Each of 4 spinor components carries different quantum numbers of mo-mentum (p z ), Lorentz spin ( Ŝz ), and the chirality (L or R, which is determined by helicity ĥ = p • Ŝ = − or +, in the massless case), shown in Table I.
We summarize how C, P, and T act on the spinor and its various quantum numbers intuitively in Table II:  • The unitary C switches between the particle ⇔ antiparticle, but keeps the momentum p z , the spins Ŝz , and the chirality intact.Note that the antiparticle's 1st, 2nd, 3rd, 4th components of the 4-component spinor have the quantum numbers of the Ŝz and chirality (opposite with respect to those of the particle's): (−, +, −, +) for Ŝz , and (R, R, L, L) for chirality.See various clarifications in [26].
• The unitary P switches between the momentum p z > 0 ⇔ p z < 0, also switches between the chirality L ⇔ R, but keeps the spin Ŝz intact.
• The antiunitary T switches between the momentum p z > 0 ⇔ p z < 0 and the spin Ŝz 's ↑⇔↓, but keeps the chirality intact.
Below we manifest the C-P-T transformation of Table II explicitly in a set of gamma matrices acting on the spinor ψ.We adopt the standard Pauli matrix convention for the gamma matrices of Clifford algebra {γ µ , γ ν } = 2g µν with the metric signature (+, −, −, −) in the chiral Weyl basis: The active C-P-T transformation on the fields changes ψ to ψ (instead of the passive transformation on coordinates), but we adopt the primed coordinate notations, x P and x T , introduced earlier in eq. ( 1): The unitary C says C(zψ(x))C −1 = z(−iγ 2 ψ * (x)) with a linear map on a complex number z ∈ C. The C in eq. ( 5) indeed agrees with Table II, by taking into account that the spin ( Ŝz ) and chirality (L/R) quantum numbers of anti-particle ψ * are opposite to that of particle ψ in Table I, namely (−, +, −, +) and (R, R, L, L) for each of four components of spinor ψ * .
Clearly the Dirac spinor theory (here d + 1 = 3 + 1) action d d+1 x ψ( i γ µ ∂ µ − m)ψ, preserves the discrete symmetry transformations in eq. ( 5).Lo and behold, based on a chain of remarks listed below eq.( 6), we discover the total discrete nonabelian finite group structure, of C/P/T and (−1) F , summarized as Gψ ≡ Let us now elaborate on eq. ( 6) in detail step-by-step: 1. We have T 2 = (−1) F so the time-reversal Z T 2 and fermion parity Z F 2 combines to be an order-4 abelian group Z TF 4 ⊃ Z F 2 such that the total group Z TF 4 sits in the group extension of the quotient Z F 2 extended by the normal subgroup Z F 2 , written as a short exact sequence: 2. Remarkably CP = (−1) F PC here, while we can show CPψP −1 C −1 = −iγ 2 γ 0 ψ * (x P ) and PCψC −1 P −1 = + iγ 2 γ 0 ψ * (x P ) in this particular basis.This means the C and P do not commute in the fermion parity odd (−1) F = −1 sector (illustrated in Fig. 1), but they commute in the bosonic (−1) F = +1 sector.The C and P form a nonabelian finite group of order-8, a dihedral group D 8 , denoted by a standard group theory notation via enlisting its generators (on the left) and their multiplicative properties (on the right): Note that (CP) 2 = T 2 = (−1) F .
3. The eq. ( 6)'s vertical and horizontal group extensions are already explained in eq. ( 7) and eq.( 9) as two short exact sequences.

c = " > A A
x Z e X S f u y a l 9 V a / e 1 c v 0 2 j 6 M I j s E J q A A b X I M 6 u A N N 0 A I Y Z O A Z v I I 3 4 8 l 4 M d 6 N j 3 l r w c h n D s E f G J 8 / y r q V 5 g = = < / l a t e x i t > 0 C ( x) < l a t e x i t s h a 1 _ b a s e 6 4 = " 4 FIG. 1. Schematic illustrations (a) CP and PC act on a local Dirac fermionic excitation, two final differed by (−1) F due to CP = (−1) F PC. Namely, the following two procedures differed by a (−1) sign for a Dirac fermion: (i) Apply P map the particle to its mirror partner, then apply C to map the particle to its antiparticle.(ii) Apply C to map the particle to its antiparticle, then apply P to map the antiparticle to its mirror partner.More generally, the parity P here (in even spacetime dimensions) can be replaced by the reflection R. The P or R transformation is with respect to the origin (the black dot).The white planes indicate the spatial dimensions.The ψ C and ψ are fermionic particle and anti-particle excitation creation operators respectively.The convex or concave cusps represent the particle or hole excitations.(b) A consecutive procedure CPCP = (−1) F gives a minus sign to a fermionic excitation.
(the semi-direct product mod out the common normal subgroup N is denoted as " N ")?In Minkowski signature flat spacetime, we have G spacetime = Pin(d, 1), which not only is a double-cover of O(d, 1), but also contains a normal subgroup Spin(d, 1).All these Pin(d, 1), O(d, 1), and Spin(d, 1) sit inside the group extension: Note that a special orthogonal SO(d, 1) contains two components (π 0 (SO(d, 1)) = Z 2 ), the proper orthochronous Lorentz group SO + (d, 1) and another component that can be switched via the simultaneous R and T (say Z RT  2 ).Thus, Note that here we choose the Pin(d, 1) instead of Pin(1, d) because a generic non-isomorphism Pin(d, 1) ∼ = Pin(1, d), while the former has their T 2 and Clifford algebra as [8] T 2 = (−1) F , Cliff d,1 : e 2 0 = −1, e 2 j = 1, with j = 1, . . ., d, the later has a different property, not we required here: In short, Pin(d, 1) not only contains the Z F 2 center, but also contains four connected components, i.e., π 0 (Pin (d, 1) disconnected from each other flipped by Z R 2 and Z T 2 .6.The three discrete subgroups, Z R 2 , Z T 2 , and Z F 2 are found as some normal subgroup or quotient group in eq. ( 12).But where is the missing charge conjugation Z C 2 ?• In general, the charge conjugation is better defined mathematically [8] as a new element of the extended group in the CRT theorem, acting by conjugate linear (antilinear ) maps on the Hilbert space of statevectors.This follows Wigner's theorem on symmetries of a quantum system [27]: any transformation of projective Hilbert space that preserves the absolute value of the inner products can be represented by a linear or antilinear transformation of Hilbert space, which is unique up to a phase factor.
• In a particular narrow-minded purpose here, we can include naturally the internal symmetry G internal = U(1) into the full spacetime-internal symmetry of Dirac theory's G spacetime -internal = Pin(d, 1) Z F 2 U(1) in eq. ( 11) such that the charge conjugation C is the complex conjugation of the U(1), which maps g = e i qθ ∈ U(1) to g * = e − i qθ ∈ U(1).Thus the charge conjugation generates the outer automorphism of the U(1): Out(U(1)) = Z C 2 .In 3+1d, the outer automorphism of G spacetime -internal still is: Out(Pin(3, 1) Z F 2 U(1)) = Z 2 , the only natural charge conjugation available.
• In summary of the above, we put four Z 2 groups together: Z P 2 , Z R 2 , Z T 2 into disconnected components of eq. ( 12), and the Z C 2 can be introduced either (1) generally by a conjugate linear map on the Hilbert space of statevectors, or (2) narrowly by an outer automorphism of G internal or G spacetime -internal .Then, the order-16 group can be fitted into both eq.( 6) and eq.( 12)'s framework.
• We can also view the Gψ ≡ Then the spin-0 boson φ sits at an (anti )linear representation of G φ , but the spin-1/2 Dirac fermion ψ sits at a projective representation of G φ .The ψ carries fractional quantum numbers of G φ is in fact in an (anti )linear representation of Gψ .The spinor ψ is thus a fractionalization of a scalar φ.The symmetry extension [31] as 1 → Z F 2 → Gψ → G φ → 1 implies that whether ψ may or may not have 't Hooft anomaly in G φ , but ψ can become anomaly-free via the pullback to Gψ .
• The AI case has T 2 = +1, so we replace eq. ( 6)'s Z TF 4 by another subgroup Z F 2 × Z T 2 instead.Then eq. ( 6) reduces to a different order-16 nonabelian Gψ = D F,CP 8 × Z T 2 .• The AIII case has a subtle U(1) and T relation given by eq. ( 17), e.g., one can realize this new T as the combined T = CT [19,20] of eq. ( 5).We leave this and other symmetry realizations in upcoming works [33].
the Z C 2 acts trivially as an identity on Majorana spinor.Therefore, we shall reduce the total group structure to P-R-T-(−1) F without C. Then eq. ( 6)'s total group Gψ reduces to an order-8 abelian group, Z P 2 × Z TF 4 for the AII case, and Z F 2 × Z P 2 × Z T 2 for the AI case.
1+1D SPIN-1/2 FERMIONIC SPINORS Now we move on to the C-P-R-T fractionalization structure for 1+1d relativistic fermions.Dirac fermion: We can regard a 1+1d massless Dirac spinor ψ as two complex Weyl spinors in 1 L ⊕ 1 R (left L + right R) rep, easily seen in the Weyl basis gamma matrices: The active C-P-T transformation on ψ gives: • Remarkably CP = (−1) F PC, so we still have eq.( 9)'s D F,CP 8 .
• Again T is anti-unitary, so precisely T(zψ(x))T −1 = z * γ 0 ψ(x T ), but luckily the complex conjugation K is not manifest in this gamma matrix basis.Since T 2 = +1, the Z TF 4 in eq. ( 6) is replaced by the Z F 2 × Z T 2 .• PT commutes with every group element, so we derive that the order-16 total group is D F,CP 8 × Z PT  2 .This particular case is within AI case in eq. ( 15), we leave other spacetime-internal symmetry realizations (e.g., AII, AIII) in upcoming works [33].
2 acts trivially as an identity on the real Majorana spinor.Then we reduce the eq.( 6)'s total group to an order-8 group 0+1D MAJORANA FERMION ZERO MODES Kitaev's fermionic chain [34] is a 1+1d nonrelativistic quantum system, hosting a Majorana zero mode on each open end of 0+1d boundary.The 0+1d low energy effective boundary action is dtχ i ∂ t χ for each 0+1d real Majorana fermion χ.There is no parity P in 0+1d, and no C for the real Majorana.When the bulk of k fermionic chains with k mod 8 = 0 are protected by symmetry, the k-boundary's zero modes are not gappable (with the dimension of Hilbert space as 2 k 2 ) as long as G is preserved due to the 't Hooft anomaly in G is classified by k ∈ Z 8 [35,36].Ref. [37][38][39][40][41][42][43] suggest that at k = 2 (or k = 2 mod 4 in general) has various supersymmetric quantum mechanical interpretations.Concretely, we follow Ref. [41], which shows this boundary can realize an extended symmetry G = D F,T 1 .The fermion parity (−1) F = 1 0 0 −1 = σ 3 and the time-reversal T = 0 − i i 0 K = σ 2 K do not commute, i.e., (−1) F T(−1) F = T −1 = −T.Also T 2 = −σ 0 = −1 and T 4 = +1.This example can be interpreted as a generalization of symmetry extension [31] (in contrast to symmetry breaking) to cancel (or trivialize) the k = 2 anomaly in G by a supersymmetry extension pullback to G [41].Supersymmetry extension means that there exists some symmetry generator (here T) such that this generator switches between bosonic |B and fermionic |F sectors, thus this generator does not commute with the fermion parity (−1) F .It can be also understood as a T-fractionalization from an order-4 abelian G = Z If we change the bulk symmetry to be protected by a G = Z TF  4 , then Ref. [41] finds that the k = 2 Majorana zero mode anomaly can be canceled (or trivialized) by a supersymmetry extension pullback to an order-16 nonabelian group G = M 16 [41].It can be also understood as a T-fractionalization from an order-4 abelian G (with T 2 = (−1) F and T 4 = +1) to M 16 (with T 4 = −1 and T 8 = +1 [37,41]).
3+1D SPIN-1 MAXWELL OR YANG-MILLS GAUGE THEORY We briefly analyze C-P-R-T group structure for the spin-1 gauge theories, pure U(1) Maxwell or SU(N) Yang-Mills (YM) theories of 3+1d actions Tr(F ∧ F ) − θ 8π 2 g 2 Tr(F ∧F ) of a 2-form field strength F = da− i ga∧a with a θ-term.We will see that generalized global symmetries [44] (i.e., 1-form symmetries G [1] that act on 1d Wilson or 't Hooft line operators in contrast to 0d point particle operators) can enrich the group structure.Follow the notations of [45], the active C-P-T transformations act on spin-1 gauge bosons in terms of 1-form gauge field, a = a µ dx µ = a 0 dt + a j dx j = (a α 0 dt + a α j dx j )T α with the real-valued four-vector component (namely a α µ ∈ R) and the hermitian Lie algebra generator (namely the hermitian conjugate T α † = T α and a real Lie structure constant The gauge field associated with a real symmetric Lie algebra (namely the complex conjugate T α * = T α ) has the upper version of the sign choices.The gauge field associated with an imaginary antisymmetric Lie algebra (namely T α * = −T α ) has the lower version of the sign choices.However, overall, we can rewrite the C-P-T symmetries on the combined a µ = a α µ T α from eq. ( 19) equivalently as: ) Other than C-P-R-T symmetries (manifest at θ = 0, π), the pure U(1) gauge theory has 1-form electric and magnetic symmetries, denoted as U(1) e  [1] (no Z C 2 due to no SU(2) outer automorphism) which fermionic/bosonic extension is studied carefully in [45] also in [32].These global symmetries C-P-R-T-G [1] are preserved kinematically at θ = 0 and π, but the gauge dynamical fates (spontaneously symmetry breaking or not) are highly constrained by their 't Hooft anomalies of higher symmetries.(These 't Hooft anomalies are firstly discovered in [44,46], later found to be captured by precise invertible topological QFTs via cobordism invariants by [45,47].Dynamical constraints of these anomalies are explored in particular by [45,48].) We leave additional analysis and other general gauge groups of gauge theories (see examples in Ref. [49] for SU(N ) YM with N > 2, and Ref. [50,51] for 2+1d) for future works [33].

APPLICATIONS
As applications, we briefly apply the above results to physical pertinent systems.

For any proposed duality between two seemingly different
QFTs, their global symmetries must be matched.So the C-P-T fractionalization provides a constraint to verify the duality.
• Dirac fermions can be in the fundamental or adjoint reps of SU(2) when coupling to SU(2) gauge fields.In the case of the fundamental rep, SU(2) ⊃ Z F 2 , so the fundamental QCD 4 obtained by gauging SU(2) reduces Gψ to Z C 2 × Z P 2 × Z T 2 .However, for the adjoint rep, SU(2) ⊃ Z F 2 , the resulting adjoint QCD 4 keeps the same order-16 Gψ .In fact, this C-P-T fractionalization Gψ can provide a constraint to verify the UV-IR duality between the UV adjoint QCD 4 theory and the IR Dirac fermion theory previously studied in [52][53][54][55].
• For Dirac fermions coupled to SU(3) in the fundamental rep (which SU(3) ⊃ Z F 2 ), the resulting real-world SU(3) QCD 4 indeed can keep this C-P-T fractionalization order-16 Gψ .Moreover, the CPT theorem and Vafa-Witten theorem [56] say that CPT and P cannot be spontaneously broken in a vector-like QCD theory.If the strong CP problem further indicates that the CP (thus T) is not violated in the real-world QCD 4 (namely, say θ = 0 for the θ-term θ 8π 2 g 2 Tr(F ∧ F )), then all discrete C-P-T are preserved which implies that the order-16 Gψ can be preserved in the vacuum of the real-world QCD 4 , at least within the strong force sector.
Of course, the weak force sector breaks P and CP, so Gψ is still violated within the full Standard Model.

FRACTIONAL SPIN-STATISTICS AND CPT
Since the early studies by Pauli [57], and by Schwinger-Pauli-Lüder [1][2][3][4][5][6], physicists are intrigued by the subtle relation between the spin-statistics theorem and the CPT theorem.Some observations and comments are in order: • We were well-informed that quantum excitations in 2+1d, called anyons, can have the fractional spin s (self-statistics gives a Berry phase e i 2πs ) and also abelian or nonabelian statistics (mutual statistics), see the reviews [58,59].
• Fractional C-P-T symmetry does not necessarily imply fractional spin-statistics of anyons beyond fermions.For example, the 3+1d Dirac spinor of eq. ( 6) and eq.(14) shows that the fermion ψ sits in the projective rep of G φ and carries fractionalized C-P-T quantum numbers of G φ , but ψ sits in the (anti)linear rep of Gψ .The ψ does not have anyonic statistics, but only has fermionic statistics (spin s = 1/2, but still fractionalized with respect to a bosonic integer spin).
• Vice versa, fractional spin-statistics of anyons do not imply a fractional C-P-T symmetry, because intrinsic topological orders (that give rise to anyons) do not necessarily require any global symmetry.
• The spin-statistics theorem colloquially says the self-braiding statistics of an excitation can be deformed to the mutual-braiding statistics between two (or more) excitations, illustrated by Dirac Belt and Feynman Plate tricks [66].Thus this theorem reveals the topological properties of matter: the topological links of worldtrajectories of (semiclassical or entangled quantum) matter excitations inside the spacetime manifold.
• The CPT or CRT theorem colloquially says that our physical laws are also obeyed by a CRT image of our universe.Thus this theorem reveals the topological properties of spacetime, the disconnected components of the spacetime symmetry groups, and how the matter-antimatter are transformed under those discrete symmetries.
• We propose that the relation between the spin-statistics theorem and the CPT theorem may also shed light on the relation between the fractional spin-statistics and the fractionalized C-P-R-T structure.Follow the promise of the fractional spin-statistics studies in the past decades [58,59], we anticipate that the fractional C-P-R-T topic presented here will also offer various future applications, both relativistic or nonrelativistic, in high-energy physics or quantum material systems.
Acknowledgments -JW thanks Pierre Deligne, Dan Freed, Jun Hou Fung, Ryan Thorngren, and is especially grateful to Pavel Putrov, Zheyan Wan, Shing-Tung Yau, Yi-Zhuang You, Yunqin Zheng, and Martin Zirnbauer for helpful comments.JW appreciates Professor Yau for persistently raising the question: "Can C-P-T symmetries be fractionalized more than Z -involutions?"JW also thanks Abhishodh Prakash for the past collaborations on the fractionalization of time-reversal T symmetry in [40,41].This work is supported by Harvard University CMSA.Fractional Spin-Statistics and CPT 7

CONTENTS
w 8 u I I 6 3 E E D W s A A 4 R l e 4 c 1 5 d F 6 c d + d j E S 0 4 + c w x / I H z + Q O r 5 4 z b < / l a t e x i t > P < l a t e x i t s h a 1 _ b a s e 6 4 = " p + 0 n A F e C X g e / H I 4 R N H X e n E f n x X l 3 P p a t O S e b O Y U / c D 5 / A J g z j M 4 = < / l a t e x i t > C < l a t e x i t s h a 1 _ b a s e 6 4 = " p + 0 n A F e C X g e / H I 4 R N H w 8 u I I 6 3 E E D W s A A 4 R l e 4 c 1 5 d F 6 c d + d j E S 0 4 + c w x / I H z + Q O r 5 4 z b < / l a t e x i t > P < l a t e x i t s h a 1 _ b a s e 6 4 = " a u 6 t z 7 m r T k r m y m C P 7 A + f w A S N Z M d < / l a t e x i t > gain ( 1) F < l a t e x i t s h a 1 _ b a s e 6 4 = " 4 r H y u e E G / / b W E b k T H 8 b 4 U C s e 9 i O h K 1 R 4 q v 6 c S E k g 5 S h w d W d A 1 E D O e x P x P 6 + TK P / c S V k Y J w p C O l v k J x y r C E 8 C w R 4 T Q B U f a U K o Y P p W T A d E E K p 0 b A U d g j 3 / 8 l / S r F b s 0 8 r J T b V Y u 8 j i y K N 9 d I D K y E Z n q I a u U R 0 1 E E U P 6 A m 9 o F f j 0 X g 2 3 o z 3 W W v O y G Z 2 0 S 8Y H 9 8 6 d 5 X 4 < / l a t e x i t > di↵ered by ( 1) F < l a t e x i t s h a 1 _ b a s e 6 4 = " b G + x k W l j / h l I o + 6 O u I N l G V I 5 4 4

8 =
their inclusion notations ( →) suggest.The Gψ is the central product between D F,CP 8 × Z TF 4 mod out their common Z F 2 center subgroup, as their Z F 2 is identical.Amusingly this Gψ is isomorphic to the 16-element rank-2 matrix group known as Pauli group ≡ σ 1 , σ 2 , σ 3 generated by Pauli matrices that act on the 2-dimensional Hilbert space of 1 qubit.4. Now we show Gψ ≡ basically says two facts: (1) the first group generated by C, P, T, and the second group generated by CP, PT, CT, and T, are exactly the same order-16 nonabelian group, (2) an order-8 quaternion group Q F,CP,PT CP, PT, CT|(CP) 2 = (PT) 2 = (CT) 2 = (−1) F (10) is generated by i = CP, j = PT, and k = CT via a standard notation Q 8 = i, j, k|i 2 = j 2 = k 2 = ijk = −1 . 5. Because the Dirac spinor ψ sits in the complex 2 L ⊕ 2 R rep of spacetime symmetry Spin(3,1), we can ask: How does the order-16 nonabelian finite group fit into the Dirac theory's spacetime-internal symmetry group and enumerating the solutions of G spacetime -internal , based on Minkowski or Euclidean notations:

8 .
Majorana fermion: Other than the Dirac spinor ψ discussed above, we can ask what happens to Majorana spinor?Once we impose the Majorana condition Cψ

8 = Z T 4 Z F 2 .
The 2dimensional Hilbert space H = {|B , |F } = H B ⊕H F has a bosonic and a fermionic ground state, say |B = 1 0 and |F = 0

TABLE II
. Agree with eq.(5), we show whether each spinor component and its quantum numbers are switched under the C-P-R-T transformation.The top horizontal row shows which quantum numbers, and the left vertical column shows how C, P/R, or T acts.The "Yes" entry in the table means the discrete symmetry switches the quantum numbers.The empty entry means the quantum number is preserved.
The standard notation of the inclusion " → in G sub → G means that G contains G sub as a subgroup.This order-16 nonabelian finite group Gψ ≡