Discrete symmetries in the Kaluza-Klein-like theories

In theories of the Kaluza-Klein kind there are spins or total angular moments in higher dimensions which manifest as charges in the observable $d=(3+1)$. The charge conjugation requirement, if following the prescription in ($3+1$), would transform any particle state out of the Dirac sea into the hole in the Dirac sea, which manifests as an anti-particle having all the spin degrees of freedom in $d$, except $S^{03}$, the same as the corresponding particle state. This is in contradiction with what we observe for the anti-particle. In this paper we redefine the discrete symmetries so that we stay within the subgroups of the starting group of symmetries, while we require that the angular moments in higher dimensions manifest as charges in $d=(3+1)$. We pay attention on spaces with even $d$.

• The definition of the discrete symmetries in the (3 + 1) dimensions after letting a series or rather a group of Killing transformations to manifest the corresponding Noether's charges in (3 + 1), we shall denote these symmetries by C N , P N and T N , which means that we analyse the type of symmetries in the extra dimensional space leading to observed symmetries in (3 + 1).
There are two special examples of spaces with extra dimensions to the observed (3+ 1) on which we discuss the here proposed discrete symmetries: I. The space of M 5+1 which breaks into M 3+1 ×M 2 with M 2 which due to the zweibein compactifies in an almost S 2 [29]. Both, spin connections and vielbeins, have the rotational invariance around the axes perpendicular to the M 2 surface, manifesting correspondingly the U (1) charge in d = (3 + 1).
ii. The space of M 13+1 which breaks into M 3+1 × the rest [30], manifesting again rotational symmetries responsible for the charges in d = (3 + 1), required by the standard model. There are vielbein and spin connection fields in d > 4 which manifest in d = (3 + 1) as the corresponding gauge vector (and scalar [5,6]) fields after the compactification.
There are several papers [15] discussing discrete symmetries in higher dimensional spaces in several contexts. Authors discuss mostly only the parity symmetry, some of them the charge conjugation and very rare all the three symmetries. All discussions on discrete symmetries concern In this paper, sect. III, we modify the d-dimensional discrete symmetries, for example the charge conjugation operator C H (Eq. (1)) as it would follow from the (3 + 1) case by analogy, so that they work effectively in the (3 + 1) dimensional theory. As we shall see below, the connection between the effective three dimensional ones (Eqs. (17,20)), C N , T N and P , is a multiplication with products of representatives of the Lorentz group corresponding to reflections and a parity operator in higher dimensions. Our notation is that we put index H on discrete symmetries P (d−1) H , T H , and C H for the whole space, i.e.
d dimensions, while we use N for the effective discrete symmetries in only our (3 + 1) dimensions.
We define three kinds of the charge conjugation operator: C (H,N ) , C (H,N ) , and C (H,N ) . The first one operates on the single particle state, put on the top of the Dirac sea, transforming the positive energy state into the corresponding negative energy state (Eqs. (1), (17)). The second one does the job of the first one emptying [5,7] (Eqs. (5), (8), (17), (20)) in addition the negative energy state, creating correspondingly a hole, which manifests as a positive energy anti-particle state, put on the top of the Dirac sea. (The corresponding single anti-particle state must also solve the equations of motion as the starting particle state does, although we must understand it as a hole in the Dirac sea in the context of the Fock space). The third one Eqs. ((5), (17)) is the operator, operating on the second quantized state (Eq.(3)).
Discrete symmetries presented in this paper commute with the family quantum numbers -the family groups defining the equivalent representations with respect to the spin and correspondingly to all the charge groups have no influence on the here presented discrete symmetries [34].
Although we illustrate our proposed discrete symmetries in two special cases (sects. II A, III A, III B, IV), in which fermions, sect. III B, interact in the Kaluza-Klein way with the vielbein and spin connection fields, the proposed redefinition of the discrete symmetries, marked by index N , is expected to be quite general, offering experimentally observed properties of anti-particles in d = (3 + 1) for the Kaluza-Klein kind of theories, helping also to define the discrete symmetries in Our effective parity, P d−1 N , Eqs. (17,20), proposal does, however, not contain any transformation of the extra dimensional coordinates and just got the contribution of the γ a matrices adjusted so that the extra dimensional gamma matrices γ 5 , γ 6 , ..., γ d−1 ,γ d commute with P d−1 N . This means that this operation is quite insensitive to the extra dimensions in such a way that it is not important if the extra dimensional space obeys any parity like symmetry.
We pay attention on spaces with even d [35].
We do not discuss the way how does an (almost) compactification happen in our here discussed two particular cases. In the ref. [14] we propose vielbein and spin connection fields which are responsible for the compactification of an infinite surface into an almost S 2 , but do not tell what (fermion condensates) causes the appearance of these gauge fields. These studies are for the two cases, presented in this paper, under consideration. There are, however, several proposals in the literature which suggest the compactification scheme and discuss it [16]. We are not yet able to comment them from the point of view of our two discussed cases.
We discuss the generality of our effective proposal for discrete symmetries in section IV, in subsection IV A of which we discuss our two special cases, commenting also possible way of compactifying the higher dimensional space.
We shall use the concept of the Dirac sea second quantized picture, which is equivalent to the formal ordinary second quantization, because it offers, in our opinion, a nice physical understanding.
We do not study in this paper the break of the CP and correspondingly of the T symmetry.

II. DISCRETE SYMMETRIES IN D-DIMENSIONS FOLLOWING THE DEFINITIONS
We start with the definition of the discrete symmetries as they follow from the prescription in d = (3 + 1). We treat particles which carry in d dimensions only spin, no charges. They also carry the family quantum numbers, which, however, commute with the discrete family operators.
We first treat free spinors. We define the C H operator to be distinguished from the C H operator.
The first transforms any single particle state Ψ pos p , index p denotes the fermion state, which solves the Weyl equation for a free massless spinor with a positive energy and it is in the second quantized theory understood as the state above the Dirac sea, into the charge conjugate one with the negative energy Ψ neg p and correspondingly belonging to a state in the Dirac sea The product of the imaginary γ a operators is meant in the ascending order. We make a choice of γ 0 , γ 1 real, γ 2 imaginary, γ 3 real, γ 5 imaginary, γ 6 real, and alternating real and imaginary ones we end up in even dimensional spaces with real γ d . K makes complex conjugation, transforming i into −i.
We define C H as the operator, which emptyies the negative energy state in the Dirac sea following from the starting positive energy state, and creates an anti-particle with the positive energy and all the properties of the starting single particle state above the Dirac sea -that is with the same dmomentum and all the spin degrees of freedom the same, except the S 03 value, as the starting single particle state. The operator S 03 is involved in the boost (contributing in d = (3 + 1), together with the spin, to handedness) and does not determine the (ordinary) spin. Accordingly we do not have to keep the S 03 value a priori unchanged under the charge conjugation. Had we instead considered CP we would also have kept S 03 .
Let Ψ † p [Ψ pos p ] be the creation operator creating a fermion in the state Ψ pos p (which is a function of x) and let Ψ p ( x) be the second quantized field creating a fermion at position x. Then or on a vacuum where it describes a single particle in the state Ψ pos so that the anti-particle state becomes We also can derive the relation This formal operation C Hf ormal means the action on the second quantized field Ψ as if it were a function of x and a column in gamma matrix space, and that the complex conjugation is replaced by the Hermitian conjugation ( † ) on the second quantized operator [36].
Let us define the operator "emptying" [4,7] the Dirac sea, so that operation of "emptying" after the charge conjugation C H (which transforms the state put on the top of the Dirac sea into the corresponding negative energy state) creates the anti-particle state to the starting particle state, both put on the top of the Dirac sea and both solving the Weyl equation for a free massless although we must keep in mind that indeed the anti-particle state is a hole in the Dirac sea from the Fock space point of view. The operator "emptying" is bringing the single particle operator C H into the operator on the Fock space. Then the anti-particle state creation operator -Ψ † a [Ψ pos p ] -to the corresponding particle state creation operator -can be obtained also as follows The operator C H = "emptying" · C H operating on Ψ pos p ( x) transforms the positive energy spinor state (which solves the Weyl equation for a massless free spinor) put on the top of the Dirac sea into the positive energy anti-spinor state, which again solves the Weyl equation for a massless free anti-spinor put on the top of the Dirac sea. Let us point out that the operator "emptying" transforms the single particle operator C H into the operator operating in the Fock space.
The operator "emptying" operates meaningfully in all known cases when the higher dimensions manifest charges or masses or both in d = (3 + 1) space.
We define the time reversal operator T H and the parity operator P (d−1) H as follows Again the product γ a is meant in the ascending order in γ a .
Let us calculate now the product of C H P with ∝ stays for up to a phase. It follows and show the application of the above defined discrete symmetries on the solutions for two particular cases: i. d = (5 + 1), the properties of which we study in several papers [2,14], and ii. d = (13 + 1), which one of the authors of this paper uses in her spin-charge-family theory [5,6,[17][18][19][21][22][23], since it manifests in d = (3 + 1) in the low energy regime the family members (explaining correspondingly the appearance of families) with the family members assumed by the standard model (extended with the right handed neutrino). Let us recognize that the operator of handedness, expressed in terms of the Cartan subalgebra members, is as follows For the choice of the coordinate system so that d-momentum manifests p a = (p 0 , 0, 0, p 3 , 0 . . . 0) the Weyl equation simplifies to We shall make use of this choice. Solutions in the coordinate representation are plane waves: e −ip a xa . In this part T H and P H manifest as follows (+) e −ip 0 x 0 +ip 3 x 3 , for example. We use the technique of the refs. [25,26]. A short overview can be found in the Appendix. The reader is kindly asked to look for more detailed explanation in [26]. It follows for p 0 = |p 0 | and p 3 = |p 3 | This state is the solution of the Weyl equation for the negative energy state. But the hole of this state in the Dirac sea makes a positive energy state (above the Dirac sea) with the properties of the starting state, but it is an anti-particle state: Ψ pos a1 = 03 (+i) 12 (+)

56
(+) e −ip 0 x 0 +ip 3 x 3 , defined [37] on the Dirac sea with the hole belonging to the negative energy single-particle state ψ neg 2 . Namely, C H Ψ[Ψ pos p ] C −1 H , when applied on the vacuum state, represents an anti-particle.
This anti-particle state is correspondingly the solution of the same Weyl equation, and it belongs to the same representation as the starting state (and C H is obviously a good symmetry in this d = 2 ( mod 4) space). The operator C H from Eq. (8), applied on the state ψ pos p1 , gives the same result: ψ pos a1 , which belong to the same representation of the Weyl equation as the starting state. But this state has the S 56 spin, which should represent in d = (3+1) the charge of the anti-particle, the same as the starting state. This is not in agreement with what we observe. e −ip 0 x 0 +ip 3 x 3 = ψ pos 2 , and solving the Weyl equation. Also the product of all three discrete symmetries is correspondingly a good symmetry as well, transforming the starting state (put on the top of the Dirac sea) into the positive energy anti- , which is the hole in the state ψ neg  Let us now look at d = (13 + 1) case, the positive energy states of which are presented in Table II.
Following the procedure used in the previous case of d = (5 + 1), the operator C H transforms, let say the first state in Table II, which represents due to its quantum numbers the right handed (with respect to d = (3 + 1)) u-quark with spin up, weak chargeless, carrying the colour charge ( 1 2 , 1 (2 √ 3) ), the third component of the second SU (2) II charge 1 2 , the hyper charge 2 3 and the electromagnetic charge 2 3 , while it carries the momentum p a = (p 0 , 0, 0, p 3 , 0, . . . , 0), as follows [−] || 9 10 [−] 11 12 [+] 13 14 [ This state solves the Weyl equation for the negative energy and inverse momentum, carrying all the eigenvalues of the Cartan subalgebra operators (S 12 , S 56 , S 78 , S 9 10 , S 11 12 , S 13 14 ), except S 03 , of the opposite values than the starting state (this negative energy state is a part of the starting Weyl representation, not presented in Table II, but the reader can find this state in the ref. [18]). The second quantized charge conjugation operator C H empties C H u 1R in the Dirac sea, creating the anti-particle state to the starting state with all the quantum numbers of the starting state, obviously in contradiction with the observations, that the anti-particle state has the same momentum in d = (3 + 1) but opposite charges than the starting state.
We look for new discrete symmetries, which would lead to the desired properties of the anti-particle state to any second quantized state: i. The anti-particle state has the same momentum in d = (3 + 1) as the starting state.
ii. The anti-particle state has the opposite values of the Cartan subalgebra of the total angular momentum J st = L st + S st , (s, t) ∈ (5, 6, . . . , d) (or at low energies rather the opposite values of the Cartan subalgebra of S st , (s, t) ∈ (5, 6, . . . , d)) as the starting state.
The manifestation of the total angular momentum (in the low energy regime rather the spin degrees of freedom) in d > 4 as charges in d ≤ 4 depends on the symmetries that (non-)compact spaces manifest [14]. (For the toy model [14] in d = (5 + 1) the spin on the infinite surface, curled into an almost sphere, manifests for a massless spinor as a charge in d = (3 + 1). Only to the massive states the total angular momentum in d = (5,6) contributes.) In the case of the spincharge-family theory in d = (13 + 1), which manifests at low energies properties of the standard model, the operators τ 1 , τ 2 , τ 3 , Y, τ 4 , Q, or rather their superposition (which all are superposition of S ab , a, b ∈ {5, 6, . . . , 14}) define the conserved charges in d = (3+1) before and after the electroweak break.
We define new discrete symmetries by transforming the above defined discrete symmetries (C H , C H , C H , T H , P H ) so that, while remaining within the same groups of symmetries, the redefined discrete symmetries manifest the experimentally acceptable properties in d = (3 + 1), which is of the essential importance for all the Kaluza-Klein theories [9][10][11][12] without any degrees of freedom of fermions besides the spin and family quantum numbers [6,17]. We define new discrete symmetries as follows 5 6 e iπJ 7 8 e iπJ 9 10 e iπJ 11 12 e iπJ 13 14 , . . . , e iπJ (d−1)d , The operator for "emptying" is defined in Eq.(4) as "emptying" = ℜγ a γ a K, the operator C N = ℜγ a γ a K C N , while the operator C N is defined according to Eq. (5) as The rotations (e iπJ 1 2 e iπJ 3 5 e iπJ 7 9 . . . , e iπJ (d−3)(d−1) ) together with (multiplied by) P All three new operators commute among themselves as also the old ones do. The shorter expressions for the same discrete operators of Eq. (17) are up to a phase Operators I operates as follows: The above defined operators C H , P is not a good symmetry.
Let us make the charge conjugation operation C N on the second quantized state Ψ † [u 1R ], the corresponding single-particle state of which, put on the top of the Dirac sea, is presented in the first line of [−] 11 12 [+] 13 14 [ To apply C N on u 1R we must, according to the definition in the first line of Eq. (17), multiply The corresponding second quantized state is the hole in this single particle negative energy state in the Dirac sea (Fock space), which solves the Weyl equation for the negative energy state. It is the state [−] || 9 10 [−] 11 12 [+] 13 14 [+] e −ip 0 x 0 +ip 3 x 3 . [−] 11 12 [+] 13 14 [+] e −ip 0 x 0 +ip 3 x 3 |vac > f am ) has the right charges, that is the opposite ones to those of the corresponding particle state, it is not a good symmetry. Again this is not within the same Weyl representation and correspondingly C N is not a good symmetry in d = (13 + 1).
In all the spaces with d = 2 ( mod 4) the charge conjugation operator C N is not a good symmetry within one Weyl representation: With a product of an odd number of γ a it jumps out of the starting Weyl representation.
Parity symmetry P In d = (5 + 1) we apply C N P as follows: (Table I), which must be put on the top of the Dirac sea, representing the hole in the state ψ neg [−] 11 12 [+] 13 14 [+] e −ip 0 x 0 −ip 3 x 3 . This state (which solves the Weyl equation γ a p a Ψ = 0) gives, put on the top of the Dirac sea, the corresponding antiparticle, belonging to the same Weyl representation, and it is left handed with respect d = (3 + 1).
This anti-particle is recognized as a left handed weak chargeless anti u-quark, of the anti-colour charge, belonging to the same Weyl representation (see the ref. [18], Table 4., line 35).
Following Eq. (9), the creation operator for an anti-particle state, which is C N P (d−1) N transformed creation operator for the particle state is therefore  eigenvalue and opposite p 3 and S 12 . Obviously T N is a good symmetry.
In the case of d = (13 + 1) operator T N transforms u 1R with spin up from Table II Table I and creating the particle state, into the creation operator for the positive energy anti-particle state This state has an opposite handedness in d = (3 + 1) and also the opposite spin and the opposite "charge".
In d = (13 + 1) the operator C N P  Table I, put on the top of the Dirac sea, into the positive energy anti-particle state with the properties ofū 1 L from the ref. [18], Table 4., line 36) (put on the top of the Dirac sea): weak chargeless, with the spin down and of the anti-colour charge T N is a good symmetry, as it is expected to be.

B. Interacting spinors
Let us assume quite a general Lagrange density for a spinor in d = ((d − 1) + 1) dimensional space, which carries, like in the Kaluza-Klein theories, the spins and no charges f α a are vielbein and ω cdα spin connection fields, the gauge fields of p a and S ab , respectively. In this paper we do not discuss the families quantum numbers, which commute with here defined with m = (0, 1, 2, 3), s = (5, 6, . . . , d).
st S st f σ s ω stσ . One finds that all in agreement with the standard knowledge for the gauge vector fields and charges in d = (3 + 1) [27].
One can check also that C N P ) , concerning in d = (3 + 1) the gauge scalar fields. The later determine massless and massive solutions for spinors and, if gaining nonzero vacuum expectation values, contribute not only to masses of spinors but also to those gauge fields, to which they couple.
There exist in (almost) compactified spaces M d−4 , for particular choices of vielbeins and spin connection fields in Eq. (25), massless and massive solutions [2,14]. In subsect. IV A we discuss such a case for d = (5 + 1). One finds that the operator C N P (d−1) N transforms either the massless or massive solutions of the Weyl equation, which represent particle states on the top of the Dirac sea, into their anti-particle states, which are holes in the Dirac sea. It follows also for the case that the infinite surface in the fifth and the sixth dimensions compactifies into an almost S 2 with the radius ρ 0 that the massive state ψ which is to be taken into account together with the C N P N T N invariance.  [41]) torus with momenta as the conserved charges would not be of our type, still our proposal might be of a help to find the definition of appropriate symmetries also for such cases.
We got the proposals for the discrete symmetries for the effective (3 + 1) theory (Eqs. (17,20)) from analysing our special case, for which one immediately sees that the proposal for P d−1 Looking at Eq. (20) one sees that the background fields have to obey some reflection symmetry in order that T N and C N be well defined symmetries. (One needs well defined discrete symmetries even if in particular cases each of them is not a good symmetry, when the handedness of spinors prevent them to be a good symmetries, while the product of the two is then a good symmetry.) So, unless the extra dimensional back ground fields obey in even d the reflection symmetry for while for T N they obey the equations of motion for spinors do not have these symmetries of T N and C N . One easily checks that the toy model [14] has the above (Eqs. (20, 29, 30)) symmetry.
These requirements for the extra dimensional reflection for background and fermion fields of Eqs. (29,30) are due to our request that anti-particles should manifest in (3 + 1) dimensions opposite charges as particles (the charges of which correspond to appropriate "Killing forms"). (So that C N inverts the charges.) One can understand the alternating reflection properties of x s , s ≥ 5, Eq. (29), in our example of the toy model [14], by the requirement that the "Killing forms", which are circles around the fixed point, must change the orientation.
Concerning the alternating reflection (in coordinate space) of T N in Eq. (30) one can understand this alternation by again looking at our example of the toy model [14]. Since T H (Eq. (6) In the torus case we need the true parity P H × P N in extra dimensions to change the signs of "Killing forms".
In complicated cases we can a priori imagine that constructing appropriate reflections inverting the signs of all the to be charges "Killing forms" could be complicated.
If the background fields are mainly just the metric tensor fields with extra dimensional components and the charges commute, it would not be difficult to find for each separate charge a reflection symmetry, reflecting just that symmetry, just that charge. Combining these reflections for the separate charges to a combined reflection reflecting all the charges would then be a proposal for the replacement for (30) and (29).
Let us mention the ref. [24] with one of the authors of this paper (H.B.N.) as a coauthor.
The book stresses that symmetries can often be derived from small assumptions which we put into a theory. For the discrete symmetries for the strong and electromagnetic interactions one ought to assume: i. Anomaly cancellations, ii. Small group representations and iii. Charge quantization rule. This author understands their derivation as a competitive way of deriving the discrete symmetries operators without knowing the theory behind.
Let us add that the Calabi-Yau kind of spaces [8] seems to have the symmetry so that our proposed discrete symmetries work.

A. Comments on two special cases
In the subsection III B we discuss how do our proposed discrete symmetries, Eq. In this subsection we shortly present the fields, zweibeins and spin connections, which in our toy model [2,14] in d = (5 + 1) cause an almost compactification. We also comment briefly our "realistic case" in d = (13 + 1) which is offering the explanation for all the charges and gauge fields of the standard model, with the families and scalar fields included, although we do not discuss in this paper the appearance of families and correspondingly a possible explanation for the Yukawa couplings [5].
In the ref. [14] we present the zweibeins and the spin connection fields, assumed to be caused with f = 1 + ( ρ 2ρ 0 ) 2 = 2 1 + cos ϑ , and the spin connection field where ρ 0 is the radius of S 2 . It follows that this choice of the spin connection field on an almost S 2 allows for 0 < 2F ≤ 1 only one normalizable (square integrable) massless solution -the left handed spinor with the Kaluza-Klein charge in d = (3 + 1) equal to 1 2 . The massless and massive solutions preserve the rotational symmetry around the axis perpendicular to the surface in the fifth and the sixth dimension and are correspondingly the eigenfunction of the total angular momentum M 56 = x 5 p 6 − x 6 p 5 + S 56 = −i ∂ ∂φ + S 56 , M 56 ψ (6) = (n + 1 2 ) ψ (6) . For the choice of the coordinate system p a = (p 0 , 0, 0, p 3 , p 5 , p 6 ) the massive solution with the Kaluza-Klein charge n + 1/2 solves the equation of motion, derived from the Lagrange function Eq. (25), with A n and B n+1 determined by the equations There exists the massless left handed spinor with the Kaluza-Klein charge in d = (3 + 1) equal to For F = 1 2 and p 1 = 0 = p 2 this solution corresponds to the particle described by ψ pos with the two functions A −(n+1) and B −n , which solve the equations where F goes to −F , in accordance with the C N P (d−1) N = γ 0 γ 5 I x 3 I x 6 transformation requirement for the fields.
One easily sees that ψ . Would the scalar (with respect to (d = (3 + 1))) f σ s ω 56σ achieve nonzero vacuum expectation values breaking the rotational symmetry on the (5, 6) surface, the charge S 56 would no longer be conserved and the scalar fields would behave similar as the Higgs of the standard model, carrying in this case the "hypercharge" S 56 .
In the case of d = (13 + 1) the compactification is again assumed to be triggered by spinor condensates which then cause the appearance of vielbeins and spin connection fields. The compactification from the symmetry SO(13, 1) (first to SO(7, 1) × U (1) II × SU (3) and then) to (3), leaving all the family members massless (in the toy model case we found the solution for the compactification of the (x 5 , x 6 ) surface into an almost S 2 for particular spin connections and vielbeins) ensure that the spins in d > 4 (in the low energy limit, otherwise the total angular momenta) manifest in d = (3 + 1) all the observed charges.
(There are in the theory [5,6,[17][18][19][21][22][23] two kinds of spin connection fields. The second one, not discussed in this paper, takes care of families. Correspondingly there are before the electroweak break four, rather than three so far observed, massless families of quarks and leptons.) We don't yet have the solution for the compactification procedure not even comparable with the one for the toy model in d = (5 + 1). This study is under consideration.
However, analysing a massless left handed representation in d = (13 + 1) -similarly as in the case of the toy model but in this case taking into account the charge groups of quarks and leptons assumed by the standard model, they are subgroups of SO(13, 1) -one easily sees that one (each) family representation in d = (13 + 1) contains [17][18][19] the left handed (with respect to d = (3 + 1)) weak charged coloured quarks and colourless leptons with particular spinor quantum number ( 1 6 for quarks and − 1 2 for leptons) and zero hyper charge and the right handed weak chargeless quarks and leptons, with the spinor charge of the left handed ones but with the hyper charges as required by the standard model. In Table II are u and d quarks of a particular colour presented, left and right handed ones. Leptons distinguish from the quarks in the colour and in the spinor quantum numbers. One can find the whole one family representation in the ref. [18] and in Table III of Appendix .
When the scalar spin connection fields of the two kinds (bringing appropriate weak and hyper charges to the right handed members of one family) gain nonzero vacuum expectation values, the electroweak break occurs, causing that the fermions and the weak bosons become massive, while the U (1) electromagnetic field stay massless.
The effective Lagrange density is presented in Eq. ( 26).
The termψγ s p 0s ψ is responsible for masses of spinors in d = (3 + 1), with γ 0 γ s , s = (7,8) transforming the right handed quarks and leptons, weak chargeless and of particular hypercharge into the left handed weak charged partners.
Similarly as in the case of the toy model the discrete symmetries of Eq. ( 20) keep their meaning also in this case.

V. CONCLUSIONS
We define in this paper the discrete symmetries, C N , P N and T N (Eqs. (17,20)) in even dimensional spaces leading in d = (3 + 1) to the experimentally observed symmetries, if the Kaluza-Klein kind of a theory [9][10][11][12] with d > (3 + 1) determining charges in d = (3 + 1) (among them also the spin-charge-family proposal of one of us (N.S.M.B. [5, 17-19, 21-23, 25] offering also the mechanism for generating families), is the right way to understand the assumptions of the standard model. We indeed define three kinds of the charge conjugation operators: Besides C N , which operates on the creation operator for a particle, also C N transforming the positive energy state representing a particle when put on the top of the Dirac sea into its negative energy partner, and C N which empties this negative energy state in the Dirac sea representing on the top of the Dirac sea the anti-particle state (18).
Although we designed this discrete symmetry operators for cases with a central point symmetry (see sect. IV) (there might be several) and particular rotational symmetries around the central point in higher dimensions, yet our proposal might help to define these discrete symmetries also in more complicated cases, as discussed in sect. IV. We do not study in this paper the break of C N , P N and T N symmetries.
We analyse properties of the proposed symmetries from the point of view of the observables in d = (3 + 1). Our definition of discrete symmetries is, as discussed in this paper and in particular in sect. IV, more general and valid for spaces with the central points and rotational symmetries around these points and might be helpful also for finding appropriate discrete symmetries operators in examples, when compactification is made by a torus, where the generators of translations around the torus are declared as charges in (3 + 1).
These discrete symmetries do not distinguish among families of fermions as long as the family groups form equivalent representations with respect to the charge groups.
We illustrate our definition of the discrete symmetries on two cases: i. d = (5 + 1) and ii. d = (13 + 1). The first case is a toy model on which we show [2,14,26] that the Kaluza-Klein kind of theories can lead in non-compact spaces to observable (almost massless) properties of fermions. We present in Table I one family of fermions of positive and negative energy states. We also presented a way for a possible compactification in this toy model to demonstrate that our definition of the discrete symmetries is meaningful IV A.
For the second illustration of the proposed discrete symmetries the one family spinor representation of the spin-charge-family theory, which explains the assumptions of the standard model, is taken. We present in Table II Table 2. line 1 and Table 4. line 35). C N P N transforms the weak charged ( 1 2 ) left handed neutrino, with spin up and colour chargeless into the right handed weak anti-charged (− 1 2 ) anti-neutrino with the spin up, anti-colour chargeless (see Appendix Table III, line 31 and 61 and also the ref. [18], Table 3, line 31 and Table 5, line 61).
We also discuss about an acceptable compactification procedure, which leads in this case to the standard model as a low energy effective theory of the spin-charge-family theory. This study is in progress.
We concentrated on discrete symmetries of fermions, but discussed also the properties of bosonic fields in higher dimensions, which are assumed to be treated as background fields, discussing in sect. IV their behaving with respect to both kinds of the discrete symmetries: C N , P  [6,[9][10][11][12]17], in which the way of curling up the higher dimensional space into (almost) compact spaces or non compact spaces do not break a parity.
To discuss discrete symmetries of Kaluza-Klein kind of theories proposed in the literature [8,15] from the point of view of our proposal would require our complete understanding of these models and in addition discussions with the authors.
Appendix: The technique for representing spinors [6,25,26], a shortened version of the one presented in [6] The technique [6,25,26] can be used to construct a spinor basis for any dimension d and any signature in an easy and transparent way. Equipped with the graphic presentation of basic states, the technique offers an elegant way to see all the quantum numbers of states with respect to the Lorentz groups, as well as transformation properties of the states under any Clifford algebra object.
The objects γ a have properties {γ a , γ b } + = 2η ab I, for any d, even or odd. I is the unit element in the Clifford algebra.
The Clifford algebra objects S ab close the algebra of the Lorentz group S ab := (i/4)(γ a γ b −γ b γ a ) , {S ab , S cd } − = i(η ad S bc + η bc S ad − η ac S bd − η bd S ac ) . The "Hermiticity" property for γ a 's: γ a † = η aa γ a is assumed in order that γ a are formally unitary, i.e. γ a † γ a = I.
The Cartan subalgebra of the algebra is chosen in even dimensional spaces as follows:   There are two kinds of the Clifford algebra objects [5,25]: besides the Dirac γ a ones alsoγ a , with the properties [25] {γ a ,γ b } + = 2η ab , {γ a ,γ b } + = 0 , (A.6) for any d, even or odd.γ a form the equivalent representations with respect to γ a . If γ a multiply any Clifford algebra object B = i=0,d a a 1 ···a i γ a 1 · · · γ a i from the left hand side (γ a B|vac > f am , |vac > f am is the vacuum state), multiplyγ a the same B from the right hand side (γ a B|vac > f am = i(−) n B Bγ a |vac > f am . (−) n B = +1, −1, when the object B has an even or odd Clifford character, respectively.