"An effective two dimensionality"cases bring a new hope to the Kaluza-Klein[like] theories

One step towards realistic Kaluza-Klein[like] theories and a loop hole through the Witten's"no-go theorem"is presented for cases which we call an effective two dimensionality cases: In $d=2$ the equations of motion following from the action with the linear curvature leave spin connections and zweibeins undetermined. We present the case of a spinor in $d=(1+5)$ compactified on a formally infinite disc with the zweibein which makes a disc curved on an almost $S^2$ and with the spin connection field which allows on such a sphere only one massless normalizable spinor state of a particular charge, which couples the spinor chirally to the corresponding Kaluza-Klein gauge field. We assume no external gauge fields. The masslessness of a spinor is achieved by the choice of a spin connection field (which breaks parity), the zweibein and the normalizability condition for spinor states, which guarantee a discrete spectrum forming the complete basis. We discuss the meaning of the hole, which manifests the noncompactness of the space.


I. INTRODUCTION
The idea of Kaluza and Klein [1] of obtaining the electromagnetism -and under the influence of their idea nowadays also the weak and colour fields [2][3][4][5][6][7][8]13] -from purely gravitational degrees of freedom connected with having extra dimensions is very elegant.
More than twenty fives years ago the Kaluza-Klein[like] theories were studied very intensively by many authors [8,9,13,14]. Although the breaking of the symmetry of the starting Lagrange density to the low energy effective ones (that is to the charges and correspondingly to the gauge fields assumed by the standard model of the electroweak and colour interactions) seem very promising, the idea of Kaluza and Klein was almost killed by the "no-go theorem" of E. Witten [15] telling that these kinds of Kaluza-Klein[like] theories with the gravitational fields only (that is with vielbeins and spin connections) have severe difficulties with obtaining massless fermions chirally coupled to the Kaluza-Klein-type gauge fields in d = 1 + 3, as required by the standard model. There were attempts to escape from the "no-go theorem" in compact extra spaces by having torsion [5], or by having an orbifold structure [11], or by putting extra gauge fields by hand in addition to gravity in higher dimensions [12], which is no longer the pure Kaluza-Klein[like] theory and loses accordingly the elegance.
Since there is the assumption that the space is compact in the "no-go theorem" of E. Witten, there are also the attempts to achieve masslessness by appropriate choices of vielbeins in noncompact spaces, one of works [13] is commented in the footnote [27].
There are several attempts to point out the importance of non compact extra dimensions, like [16], many of them surveyed in [17]. These attempts do not really try to keep the Kaluza-Klein approach in the original elegant version, they rather embed strings, membranes, pbranes into higher dimensional spaces. The most popular models of this kind are probably Randall-Sundrum models [18].
We are interested in this paper in extra dimensions in the Kaluza-Klein sense: that is as a possibility that the gravity (and only gravity) in extra dimensions manifests as the standard model gauge fields in (1 + 3), coupled to the corresponding charges. In refs. [24] we achieved masslessness of spinors in the pure Kaluza-Klein[like] theory (for the case of M 1+5 manifold broken into M 1+3 × an infinite disc) with the appropriate choice of a boundary limiting the extra dimensions on a finite surface on a disc.
In the proposed paper we take the whole two dimensional plane, and roll it up into an almost S 2 with one point -the south pole -excluded. It is our choice of a zweibein which forces the two extra dimensions into an almost S 2 . Thus, although it has a finite volume (namely the surface of S 2 ), the space is non compact. We require spinor states to be in the fifth and sixth dimensions normalizable [28], proving that the normalizable solutions form a complete set. It is our choice of a particular spin connection field, with the strengths within an interval, which allows only one normalizable massless state of a particular handedness (with respect to (1 + 3)), breaking the parity symmetry.
The finite volume of an infinite disc, an appropriate choice of the spin connection field with the strength F allowed to be within the whole interval 0 < 2F ≤ 1 and the normalizability requirement make the mass spectrum of our Hermitean Hamiltonian in a noncompact space discrete, with only one massless state of particular charge chirally coupled to the Kaluza-Klein gauge field. It is the sign of F which makes a choice of the handedness of a massless state, breaking the parity symmetry. The usually expected problem with extra non compact dimensions having a continuous spectrum is not present in our model.
For a particular choice of the strength of the spin connection field we find the states and the spectrum (the masses) analytically. This mass spectrum of states forms the complete set on our almost S 2 . For the remaining values of the strength, for all of which only one massless solution of a particular handedness in (1 + 3) exists, it is not difficult to find the recursive formulas for normalizable solutions and the masses. Accordingly in this two dimensional noncompact space, with the spin connections and vielbeins which both are a part of the gravitational gauge fields and with no presence of an (additional) external field, the "no-go theorem" of E. Witten is not valid.
We also characterize the "singularity" which the spinor solutions "feel" on our infinite disc with the zweibein of a S 2 sphere, when treating the disc as the almost S 2 sphere, that is the S 2 sphere with the hole on the southern pole, so that we have almost M (1+3) × S 2 case, that it is almost a compact space.
Let us add: As it is not difficult to recognize, the two dimensional spaces are very special [19,20]. Namely, in dimensions higher than two, when we have no fermions present and only the curvature in the first power in the Lagrange density, the spin connections are normally determined from the vielbein fields, and the torsion is zero. In the two dimensional spaces, the vielbeins do not determine the spin connection fields. In the present article we pay attention to cases, which we call an effective two-dimensionality, when the spin connections are not fully determined by the vielbeins.
In the here proposed types of models there is the chance for having chirally mass protected fermions in a theory in which the chirally protecting effective four dimensional gauge fields are true Kaluza-Klein[like] fields, the degrees of which inherit from the higher dimensional gravitational ones. We are thus hoping for a revival of true Kaluza-Klein[like] models as candidates for phenomenologically viable models! One of us has been trying for long to develop the approach unifying spins and charges and predicting families (N.S.M.B.) [21,25]  We prove in this section that in M 1+3 × an infinite disc with the particular zweibein and spin connection on the disc there exists only one massless normalizable (on the disc) fermion state of only one handedness and of a particular charge. It is accordingly mass protected.
We also present proofs that the Hamiltonian is Hermitean and the spectra of normalizable states correspondingly discrete. For a particular strength of the spin connection field we present the spectrum and states. We discuss the properties of solutions for the strengths allowed by the normalizability requirement.
Let us first repeat the four assumptions, stressed already in the introduction.
1. We assume 2(2n+1)-dimensional space, in our case n = 1, with only gravity, described by the action [30] The Riemann scalar R = R abcd η ac η bd is determined by the Riemann tensor , with vielbeins f α a [31] and the spin connections ω abα (the gauge fields of S ab = i 4 (γ a γ b − γ b γ a )). [a b] means that the antisymmetrization must be performed over the two indices a and b, E is the determinant of the inverse zweibein e s σ , e s σ f σ t δ s t , (Eq. (2)).
2. Space M 1+5 has the symmetry of M 1+3 × an infinite disc with the zweibein on the disc The last relation follows from ds 2 = e sσ e s τ dx σ dx τ = f −2 (dρ 2 + ρ 2 dφ 2 ). We use indices s, t = 5, 6 to describe the flat index in the space of an infinite plane, and σ, τ = (5), (6), to describe the Einstein index. φ determines the angle of rotations around the axis perpendicular to the disc.

The spin connection field is chosen to be
4. We require normalizability of states ψ on the disc as usual in quantum mechanics, allowing at most the plane waves normalized to the delta function: Let us make now several statements, proofs of these statements and comments, which will help to clarify the meaning of the assumptions. (1) leads to the equations of motion [19,21] {p σ , Ef } − )}ψ, n = 0, 1, 2, 3, with E = det(e a α ) = f −2 , f is from Eq.     (11,12,15)) which look for F = 1/2 like Legendre equations (Eq. (22)). It is the sign of F which makes a choice of the handedness of a massless state and breaks accordingly the parity symmetry.
One can prove that the only normalizable eigenstates in the interval 0 ≤ ρ ≤ ∞ are those with integer parameters l and n, (mρ 0 ) 2 = l(l + 1), in Eqs. (23). These states are Legendre polynomials and form the complete set. Solutions for a non integer n are singular at ρ = 0, while solutions with a non integer l are singular at ρ = ∞, both singularities make the corresponding eigenstates not normalizable. that it is the term ψ † Ef γ 0 γ s δ σ s (p 0σ + 1 2Ef {p σ , Ef } − )ψ in the Lagrange density (Eq.(6)), which manifests as the mass term m in Eq. (7). There is a term in Eq. (7), namely (6) , which clearly distinguishes between the two possible values of the spin operator S 56 in d = 5, 6, when this term applies on the state ψ (6) , dis-tinguishing correspondingly also between the two possible handedness of the state ψ (6) in . It is shown in the next subsection that a normalizable massless state (m = 0 in Eq. (7)) must fulfil the condition: ( 0 ≤ (1 − 2F 2S 56 ) < 1) ψ (6) . The sign of F chooses the handedness of a massless normalizable spinor state.
Comments 5. i.) Having the rotational symmetry around the axis perpendicular to the plane of the fifth and the sixth dimension it is meaningful to require that ψ (6) is the eigen function of the total angular momentum operator (M 56 = x 5 p 6 − x 6 p 5 + S 56 ) in the fifth and (13,14,12)). ii.) The only massless state, which fulfills the normalization condition (see Eq. (18)) for a positive F , is the state with the property 2S 56 ψ (6) = ψ (6) . Its charge (spin on the disc) is for 0 < 2F ≤ 1 equal to 1 2 as it is shown in section IV. iii.) All the other states are massive. iv.) The current in the radial direction is for all these cases equal to zero for any F . This space breaks into M 1+3 cross an infinite disc with the zweibein which formally looks almost -up to a hole in the southern pole -as a S 2 sphere, while a chosen spin connection allows on such an infinite disc only one normalizable massless state. The Hamiltonian is Hermitean, the mass spectrum of normalizable states is correspondingly discrete and the probability for a fermion to escape out of the disc is zero [33].
Allowing the whole interval of the strength of the spin connection fields (0 < 2F ≤ 1) the spin connection field is not fine tuned. For a particular choice of the constant of the spin connection field, that is for 2F = 1, the normalizable solutions are expressible with the Legendre polynomials and the massive states manifest a spectrum mρ 0 = l(l + 1), with l = 0, 1, 2, · · · and −l ≤ n ≤ 1. n + 1/2 is the charge of the spectrum. Let us point out that the "two dimensionality" can be simulated in any dimension larger than two, if vielbeins and spin connections are completely flat in all but two dimensions (this point is discussed also in the ref. [13]).

A. Solutions of the equations of motion for spinors
We look for the solutions of the equations of motion (6) for a spinor in (1+5)-dimensional space, which breaks into M (1+3) × an infinite disc curved into a noncompact "almost" S 2 sphere as a superposition of all four (2 6/2−1 ) states of a single Weyl representation. (We kindly ask the reader to see the technical details about how to write a Weyl representation in terms of the Clifford algebra objects after making a choice of the Cartan subalgebra, for which we take: S 03 , S 12 , S 56 in the refs. [25].) In our technique one spinor representation-the four states, which all are the eigenstates of the chosen Cartan subalgebra with the eigenvalues where ψ 0 is a vacuum state for the spinor state. If we write the operators of handedness in d = (1+5) as Γ (1+5) = γ 0 γ 1 γ 2 γ 3 γ 5 γ 6 (= 2 3 iS 03 S 12 S 56 ), in d = (1+3) as Γ (1+3) = −iγ 0 γ 1 γ 2 γ 3 (= 2 2 iS 03 S 12 ) and in the two dimensional space as Γ (2) = iγ 5 γ 6 (= 2S 56 ), we find that all four states are left handed with respect to Γ (1+5) , with the eigenvalue −1, the first two states are right handed and the second two states are left handed with respect to Γ (2) , with the eigenvalues 1 and −1, respectively, while the first two are left handed and the second two right handed with respect to Γ (1+3) with the eigenvalues −1 and 1, respectively. Taking into account Eq. (8) we may write the most general wave function ψ (6) obeying Eq. (7) in where A and B depend on x σ , while ψ (+) and ψ (4) (−) determine the spin and the coordinate dependent parts of the wave function ψ (6) Using ψ (6) in Eq. (7) and separating dynamics in (1+3) and on the infinite disc the following relations follow, from which we recognize the mass term m: One notices that for massless solutions (m = 0) ψ (4) (+) and ψ (4) (−) decouple. Taking the above derivation into account Eq. (7) transforms into For x (5) and x (6) from Eq. (3) and for the zweibein from Eqs. (2,3) and the spin connection Having the rotational symmetry around the axis perpendicular to the plane of the fifth and the sixth dimension we require that ψ (6) is the eigen function of the total angular momentum Accordingly we write After taking into account that S 56 Let us treat first the massless case (m = 0). Taking into account that We get correspondingly the solutions Requiring that only normalizable (square integrable) solutions are acceptable it follows for A n : −1 < n < 2F, for B n : 2F < n < 1, n is an integer.
One immediately sees that for F = 0 there is no solution for the zweibein from Eq. (3).
Eq. (19) tells us that the strength F of the spin connection field ω 56σ can make a choice between the two massless solutions A n and B n : For the only massless solution is the left handed spinor with respect to (1 + 3) If one expresses ( ρ From the above equations we see that for m = 0, that is for the massless case, the only solution with n = 0 exists, which is A , which is a constant (in agreement with our discussions above).
It is not difficult to prove that there is no normalizable solutions of Eq. (23)   , are normalizable on the infinite disc curved into almost S 2 (2 π ρdρE ψ . One can show as well that the eigenstates, with the discrete eigenvalues (ρ 0 m) 2 = l(l + 1), are orthog- for all pairs of (l, n), (l , n ), the spectrum is obviously discrete as it should be for the Hermitean Hamiltonian with the boudary conditions determined by normalizability of states.
To find solutions for all F in the interval 0 < F ≤ 1 2 , besides the massless one ψ (6)m=0 1 2 , is a more tough work. Yet one can expect that on the space of normalizable functions the Hamiltonian will stay Hermitean and since an infinitesimal change of the constant F from F = 1 2 to a tiny smaller F can not spoil the discreteness of the Hamiltonian eigenvalues, the spectrum would stay discrete. One can see that the current in the radial direction is zero for any F . We studied these solutions and found the discrete spectrum, a paper is in preparation.
(Let us recognize that e inφ P l n are spherical harmonics Y l n . Expressing ρ with ϑ, ρ 2ρ 0 = 1−cos ϑ 1+cos ϑ we rewrite the equations of motion (Eq.15)as follows

III. SINGULARITIES ON AN ALMOST S 2 SPHERE
In this section we comment on singularities "felt" by a spinor if a noncompact disc with the zweibein from Eq. (2) and the spin connections from Eq. (3) is understood as the S 2 sphere with a hole on the southern pole.
Intuitively it is not difficult to see that we are in troubles if we want the chiral fermion field of Eq. (21), that is ψ (+) , on a two dimensional space to be an eigenstate of some rotational operator M 56 , if the two dimensional space has to have the topology of S 2 , while the spin of the fermion contributes to M 56 in the "usual way" where K 56 is the Killing vector, like in Eq. (13) (K 56 = x 5 p 6 − x 6 p 5 ). Near the starting point (the origin, the northern pole of S 2 ) on the topologically S 2 sphere the Killing operator functions as the orbital angular momentum (L 56 = x (5) p (6) − x (6) p (5) ) and has to be added to the spin part S 56 , just as it is in the flat two-dimensional space. Going away from the starting point the action of M 56 may be more complicated as just a simple sum in Eq. (26).
Because of the S 2 topology there has to be namely yet another point at which the orbital Killing generator eigenvalue goes to zero, since there has to be a point, the south pole, which is left invariant under the orbital Killing transportation as it is at the starting point, at the north pole.
It is also easy to see that on the two-dimensional S 2 , the orientation of the Killing transportation in the infinitesimal neighbourhood of this second stable point, the south pole, is in the opposite direction with respect to the orientation of the Killing transportation around the north pole.
If we want to have on S 2 only a spinor of one handedness, let say the spinor ψ Therefore, embedding the S 2 sphere into a three-dimensional Euclidean space, it is not surprising that if we want a spinor of one handedness and succeed to implement it at the north pole in an outward normal direction, we can hardly implement it at the south pole.
We might hope for the compensation by the orbital part of M 56 , except at the poles. This means that we could have a state of a handed spinor if the wave function goes to zero at at least one of the poles, say the southern pole (see Eqs. (21,19)).

A. Formal introduction of a singular point
We might formally introduce at the south pole a special singularity, so that we require the wave function instead to behave at the south pole in the usual differentiable way, to be differentiable only after being multiplied (corrected) by a phase factor: Instead of ψ we require that e iφ SP ψ is our wave differentiable function in the neighbourhood of the singular point at the south pole, the phase factor e iφ SP itself behaving singularly. By making this modified requirement of the differentiability we effectively change the orbital angular momentum of the wave function by one unit ofh before we require the wave function to be smooth or differentiable. Thereby we have made the requirement that the actual wave function should have a rather unphysical extra bit of a negative angular momentum around the south pole. We must admit that it looks rather strange from the physical point of view, unless we recognize that this smoothness condition is to simulate the non-compactness of the S 2 space, which only after adding a singular point becomes an S 2 at all.
When changing the differentiability of the wave function in the neighbourhood of the singular point with the requirement that the wave function must be multiplied by a phase, we recognize that such a phase multiplication of the wave function appears when transforming From Fig. 1 we read and where x N P σ , σ = (5), (6) stay for up to now used x σ , σ = (5), (6), while x SP σ , σ = (5), (6) stay for coordinates when we put our coordinate system at the southern pole and ρ 0 is the radius of S 2 as before. We have We also can write x N P σ = ( 2ρ 0 ρ SP ) 2 (−) 1+σ x SP σ . We ought to transform the Lagrange density (Eq.(6)) expressed with respect to the coordinates at the northern pole to the corresponding Lagrange density L SP W expressed with respect to the coordinates at the southern pole by assuming We use the antisymmetric tensor ε (5)(6) = 1 = −ε (5) (6) . We recognize that The matrix O takes care that the zweibein expressed with respect to the coordinate system at the southern pole is diagonal: sin(2φ + π) − cos(2φ + π) . (32) Requiring that from where it follows that S −1 S st SO −1s s O −1t t = S s t , and recognizing that p N P In the above equation we took into account that ω N P stσ transforms into Similarly we transform the term γ s 1 The action d 2 x N P L N P W , with the density from Eq. (6), transforms, when the coordinate system is put at the southern pole, as follows which leads to the Lagrange density The requirement that S −1 γ 0 γ s S O −1t s = γ 0 γ t is fulfilled by the operator S = e −iS 56 ω 56 , and ω 56 = 2φ + π, so that in the space of the two vectors ( with φ N P = −φ SP , while we have with the property where M N P 56 = (S 56 − i ∂ ∂φ N P ) , look like when we put the coordinate system at the southern pole. When putting the coordinate system at the southern pole not only φ N P transforms into −φ SP , but also γ 6 goes into −γ 6 , accordingly When evaluating M SP 56 = (S 56 Accordingly the massless state ψ N P (6)m=0 (+) from Eq. (21) looks, when transforming the coordinate system from the northern to the southern pole, as ψ SP (6)m=0 Taking into account that x SP (5) + i2S 56 x SP (6) = ρ SP e −i2S 56 φ SP and ∂φ SP ) we can write the equations of motion as For ψ SP (6) (−) ) e −inφ SP we find the equations for When using f SP ∂ ∂ρ SP = 1 Again we find for 2F = 1 s = 5, 6; σ = (5), (6). Requiring that correspondingly the only nonzero torsion fields are those from Eq. (A.2) we find for the spin connection fields To determine the current, which couples the spinor to the Kaluza-Klein gauge fields A µ , we analyse (as in the refs. [24]) the spinor action (Eq. ( 6)) Here ψ ( We end up with the current in (1 + 3) j µ = Ed 2 xψ (6) γ m δ µ m M 56 ψ (6) . The Riemann scalar is for the vielbein of Eq. (51) equal to R = − 1 2 ρ 2 f −2 F mn F mn . If we integrate the Riemann scalar over the fifth and the sixth dimension, we get − 8π 3 (ρ 0 ) 4 F mn F mn .

V. CONCLUSIONS
We prove in this paper that one can escape from the "no-go theorem" of Witten [15], There is the discrete spectrum of normalizable eigenstates of the Hermitean Hamiltonian on the infinite disc for the chosen zweibein and spin connection of any strength F in the interval (0 < 2F ≤ 1), as we proved in section II.
The normalizable eigenstates, which are chosen to be at the same time the eigenstates of the total angular momentum on the disc M 56 = x 5 p 6 − x 6 p 5 + S 56 , with the eigenvalues (n + 1/2), carry the Kaluza-Klein charge (n + 1/2). The only massless state carries the charge ( 1 2 ). For the choice of 2F = 1 the normalizable massless state is independent of the coordinates on the disc. The normalizable massive states have the masses equal to k(k + 1)/ρ 0 , k = 1, 2, 3, .., with −k ≤ n ≤ k. The spectrum is obviously discrete and stays discrete for all F in the interval 0 < 2F ≤ 1 and for any finite ρ 0 . The current is for all the solutions and also for all F equal to zero. As long as the Hamiltonian is Hermitean on a disc, fermions can not leave the disc, unless an additional interaction (or a dynamical restoration of the symmetry M (1+5) , that is the phase transition) would force them to go out of the disc, which is not the case for our toy model. The possibility that after the break a two dimensional manifold (with the zweibein of S 2 , with one point missing and with a particular spin connection field) exists allowing only one normalizable massless state which is correspondingly mass protested and which couples to the Kaluza-Klein charge, opens, to our understanding, a new hope for the Kaluza-Klein[like] theories of the elegant version, with only the gravity, and will help to revive them.
[28] In the ref. [13], mentioned and discussed in the previous footnote, this idea of a finite volume of a noncompact space, as well as the normalizability of states is already stressed.
[29] The approach unifying spin and charges and predicting families [21] proposes in d = (1+(d−1)) a simple starting action for spinors with the two kinds of the spin generators (γ matrices): the Dirac one, which takes care of the spin and the charges, and the second one, anticommuting with the Dirac one, which generates families. For the explanation of the appearance of the two kinds of the spin generators we invite the reader to look at the refs. [21,25] and the references therein. A spinor couples in d = 1 + 13 to the vielbeins and (through two kinds of the spin generators to) the spin connection fields. Appropriate breaks of the starting symmetry lead to the left handed quarks and leptons in d = (1 + 3), which carry the weak charge while the right handed ones are weak chargeless. The approach is offering the answers to the questions about the origin of families of quarks and leptons, about the explicit values of their masses and mixing matrices (predicting the fourth family to be possibly seen at the LHC or at somewhat higher energies) as well as about the masses of the scalar and the weak gauge fields, about the dark matter candidates, and about breaking the discrete symmetries. There are many possibilities in the approach for breaking the starting symmetries to those of the standard model. These problems were studied in some crude approximations in refs. [21] and are under consideration [22].
[32] One finds that ω cda = e ω cda , if c, d, a all different while ω cda = i m ω cda , otherwise.
[33] It is expected that the zweibein curving the infinite disc into an (almost S 2 ) and the spin connection, which breaks the parity symmetry and takes a part in determining equations of motion, appear dynamically, causing the "phase transition". Accordingly could dynamical fields by causing the phase transition restore the symmetry of M 1+5 (−) e iφ ∂ ∂ϑ }Y k 0 . The expectation value of the mass operatorm on such a wave packet is k=0,∞ C k * 1/2 C k 1/2 k(k + 1)/ρ 0 .