In an e-mail discussion of his February 2007 paper "The RiemannHypothesis is a consequence of CT -invariant Quantum Mechanics",Carlos Castro said:

"... I did toy with the idea of CPT invariance. However here in reality we have a 0 + 1 dimensional theory. It is only a one-dim problem involving what I call the scaling = dilations operatord / d ln t = t d / dt I could have used a different notation by having the the symbol " x " instead of " t " . When t goes to 1/ t then the log changes sign so d/ d lnt goes to - ( d / lnt ) as it it were a reversal operation of the "time " derivative. So what counts is the dilational aspects in one-dim.

By charge conjugation C, I mean just taking the complex conjugate of the eigen-functions, eigen-states of the generalized scaling operators D_1, D_2 that are "duals" of each other. So the C operation just takes s and maps it to its complex conjugate (s)^* which is the same as taking Psi_s ( t ) to its complex conjugate since t, k, l are real-valued. Only the eigenvalue " s " is complex.

One could implement CPT invariance if one formulates the problem in 1 + 1 dimensions by introducing, for example, the hyperbolic Laplace operator in two-dimensions whose eigenfunctions are given by the modular Eisenstein' series and whose eigenvalues are also of the form s ( 1- s ). In that two-dim case, yes, you will have an honest CPT invariant theory, because besides t, you now have the variable x upon which you can apply the Parity operation x goes to - x . The Dirac opeartor and spinors make a lot of sense now in two-dim.

The key to the paper i sent you relies in adding to the scaling dilation log derivative d / d lnt, the extra terms involving the derivatives of the potential. This extra term is like adding a Gauge Field, a conection A , to the ordinary log derivative to construct a "covariant " derivative D = d + A . The Potential given by the Gauss Jacobi theta series ... is related to the Bernoulli string partition function = a Bose gas of oscillators.

To sum up the key features

- 1- Using the Scaling log derivatives d / d lnt .
- 2- Adding a potential term ... to the derivatives and with modular properties ...
- 3- Finding the eigenfunctions Psi_s ( t ) parametrized by a complex " s " variable, like in coherent states.
- 4- Exploiting the CT-invariance of the QM problem to prove why the eigenvalues s ( 1- s ) = are real.
- 5- The discrete zeros s_n are associated to the orthogonal states Psi ( t; s_n ) to the ground state related to the center of symmetry s_o = 1/2 + i 0. This leads to the proof of the Riemann Hypothesis.
Yes, it is worth looking at the 1 + 1 problem, use Dirac operators, spinors, implement CPT invariance ( instead of CT invariance ) and then perform a dimensional reduction to see what you get in 1-dim. ...".

From a Clifford Algebra point of view (see for example the book byF. Reese Harvey "Spinors and Calibrations" (Academic Press 1990) atpages 207-208:

- Cl(0,0) = R = Real Numbers with graded structure 1 corresponds to 1-dim Quantum Model used by Carlos Castro for the Riemann Hypothesis, in which the P operator is not defined so that CPT of higher dimensions becomes effectively CT.
- Cl(0,1) = C = Complex Numbers with graded structure 1 + 1 corresponds to 2-dim Feynman Checkerboard physics with CPT invariance
- Cl(0,2) = H = Quaternions with graded structure 1 + 2 + 1 corresponds to 4-dim physical spacetime with CPT invariance
- Cl(0,3) = H + H with graded structure 1 + 3 + 3 + 1 corresponds to 4-dim physical spacetime with CPT invariance plus 4-dim internal symmetry space, which is the result of introducing a preferrred quaternionic subspace into 8-dim octonionic spacetime.

Carlos Castro also said:

"... REALITY condition is the whole point in proving whyCT-invariant QM forces the spectrum s ( 1- s ) of H_A

to be REAL s ( 1- s ) = real such s = real ( location of trivialzeta zeros ) s = 1/2 + i lambda ( location of non-trivial zeta zeros). ...".

**In his February 2007 paper, The Riemann Hypothesis is aconsequence of CT -invariant Quantum Mechanics, Carlos CastroPerelman said: **

"...The Riemann's hypothesis (RH) states that the nontrivial zerosof the Riemann zeta-function are of the form s_n = 1/2 +/- ilambda_n. By constructing a continuous family of scaling-likeoperators involving the Gauss-Jacobi theta series and by invoking anovel CT -invariant Quantum Mechanics, involving a judicious chargeconjugation C and time reversal T operation, we show why the RiemannHypothesis is true. ...

Wu and Sprung ... Phys. Rev. E 48 (1993) 2595. Seealso www.maths.ex.ac.uk/~mwatkins/zeta/NTfractality.htm... have numerically shown that the lower lying non-trivialzeros can be related to the eigenvalues of a Hamiltonian whosepotential has a fractal shape and fractal dimension equal to D = 1.5.Wu and Sprung have made a very insightful and key remark pertainingthe conundrum of constructing **a one-dimensional integrable andtime-reversal quantum Hamiltonian to model the imaginary parts of thezeros of zeta as an eigenvalue problem**. This riddle of mergingchaos with integrability is solved by choosing a fractal localpotential that captures the chaotic dynamics inherent with the zetazeros. ...

The essence of the proof of the RH relies in the construction of aCT - symmetric Quantum Mechanics ... and in establishing a one to onecorrespondence among the zeta zeros s_n with the states PSI_s_n(t)orthogonal to the ground ( vacuum state ) PSI_s_o (t) associated withthe center of symmetry

of the non-trivial zeta zeros and corresponding to the fundamentalRiemann function obeying the "duality" condition Z(s) = Z( 1 - s )....

begin with the construction of the Scaling Operators related tothe Gauss-Jacobi Theta series and the Riemann zeros given by

such that its eigenvalues s are complex-valued, and itseigenfunctions are given by

D_1 is not self-adjoint since it is an operator that does notadmit an adjoint extension to the whole real line characterized bythe real variable t. The parameter k is also real-valued. ... We alsodefine the operator dual to D_1 as follows,

that is related to D_1 by the substitution t -> 1/t and bynoticing that dV(1/t)/dln(1/t) = -dV(1/t)/dlnt where V(1/t) is notequal to V(t). The eigenfunctions of the D_2 operator are

(with eigenvalue s) which can be shown to be equal to PSI_(1-s)(t)resulting from the properties of the Gauss-Jacobi theta series underthe x -> 1/x transformations. Since V(t) can be chosenarbitrarily, we choose it to be related to the Bernoulli stringspectral counting function, given by the Jacobi theta series

where w(x) = SUM{n=1, oo} e^(- pi n^2 x ) . The Gauss-Jacobiseries obeys the relation

Then, our V is such that e^(2V(t)) = G(t^l) . We defined x as t^l.We call G(x) the Gauss-Jacobi theta series (GJ)...[define]... H_A = D_2 D_1 and H_B = D_1 D_2 ...

H_B PSI_s(1/t) = s ( 1 - s ) PSI_s(1/t)

Therefore, despite that H_A, H_B are not Hermitian they have thesame spectrum s(1-s) which is real-valued only in the critical lineand in the real line. ...[the above two equations are]... theone-dimensional version of the eigenfunctions of the two-dimensionalhyperbolic Laplacian given in terms of the Eisenstein's series. HadH_A,H_B been Hermitian one would have had an immediate proof of theRH. ...

if the H_A and H_B operators are invariant under the CT operation,the RH is true ...

The invariance of the H_A,H_B operators under CT implies thevanishing commutators [H_A, CT ] = [H_B, CT ] = 0 ...When the operators H_A, H_B commute with CT , there exits neweigenfunctions PSI^CT_s (t) of the HA operator with eigenvalues s* (1- s* ). ... similar results follow for the H_B operator ... one hastwo cases to consider.

- Case A : If the pseudo-norm is null < PSI_s | CT | PSI_s > = 0 => (E_s - E*_ s ) =/= 0 ... where we have defined (E_s)* = s*(1-s*) ... then the complex eigenvalues E_s = s(1-s) and E*_s = s*(1-s*) are complex conjugates of each other. In this case the RH would be false and there are quartets of nontrivial Riemann zeta zeros given by s_n, 1-s_n, s*_n, 1-s*_n.
- Case B : If the pseudo-norm is not null : < PSI_s | CT | PSI_s > =/= 0 => (E_s - E*_s ) = 0 then the eigenvalues are real given by Es = s(1 - s) = E*_s = s*(1 - s*) ... which implies that s = real ( location of the trivial zeta zeros ) and/or s = 1/2 + i lambda ( location of the non-trivial zeta zeros). In this case the RH would be true and the non-trivial Riemann zeta zeros are given by s_n = 1/2 + i lambda_n and 1-s_n = s*_n = 1/2 - i lambda_n.

We are going to prove next why Case A does and cannot occur,therefore the RH is true because we are left with case B ... theessence of the proof relies in establishing a one to onecorrespondence among the zeta zeros s_n with the states PSI_s_n (t)orthogonal to the ground ( vacuum state ) PSI_s_o (t) associated withthe center of symmetry s_o = 1/2 + i 0 of the non-trivial zeta zerosand corresponding to the fundamental Riemann function obeying the"duality" condition Z(s) = Z(1-s).

The inner products < PSI_s_o (t) | PSI_s_n (t) > =Z[sn] = 0 fix the location of the nontrivial zeta zeros s_nsince Z[s] is proportional to zeta(s) ...

...[due to]... the consequences of ... analyticcontinuation ... of the function Z(s) to the entire complex s-plane... the construction of a genuine inner product ... of twoeigenfunctions of D_1 ... is impossible ...

The crucial problem is whether there are zeros outside thecritical line (but still inside the critical strip) and not theinterpretation ... as a genuine inner product. Despite this, we stillrather loosely refer to this mapping as a scalar product. The statesstill have a real norm squared, which however need not to bepositive-definite. Here we must emphasize that our arguments do notrely on the validity of the zeta-function regularization procedure... which precludes a rigorous interpretation ... as a scalarproduct. Instead, we can simply replace the expression "scalarproduct of PSI_s_1 and PSI_s_2" by the map S of complex numbersdefined as

(s_1, s_2) |-> S(s_1, s_2) = - ( 2/l ) Z(as + b)

where s = s*_1 + s_2 - 1/2 and a = -2/l ; b = (4k - 1)/l. In otherwords, our arguments do not rely on an evaluation of the integral ...but only on the mapping ... defined as the finite part of theintegral ... We only need to study the "orthogonality" (and symmetry)conditions with respect to the "vacuum" state so to prove why a + 2b= 1. By symmetries of the "orthogonal" states to the "vacuum" we meanalways the symmetries of the kernel of the S map.

The "inner" products are trivially divergent due to thecontribution of the n = 0 term of the GJ theta series in the integral... From now on, we denote for "inner" product ... the finite part ofthe integrals by simply removing the trivial infinity. We shall seein the next paragraphs, that this "additive" regularization is infact compatible with the symmetries of the problem. ...

if a and b are such that 2b + a = 1, then the symmetries of allthe states s orthogonal to the "vacuum" state are preserved by anymap S ... which leads to Z(as + b). In fact, if the state associatedwith the complex number s = x + iy is orthogonal to the "vacuum"state and the "scalar product" is given by Z(as+b) = Z(s'), then theRiemann zeta-function has zeros at s' = x' + iy', s'* , 1-s' and1-s'* . If we equate as + b = s', then as* + b = s'* . Now, 1-s' willbe equal to a(1 - s) + b, and 1 - s'* will be equal to a(1 - s* ) +b, if, and only if, 2b + a = 1. Therefore, all the states PSI_sorthogonal to the "vacuum" state, parameterized by the complex number1/2 + i0, will then have the same symmetry properties with respect tothe critical line as the nontrivial zeros of zeta. ...

our choice of a = -2/l and b = (4k - 1)/l is compatible with thissymmetry if k and l are related by l = 4(2k - 1). Conversely, if weassume that the orthogonal states to the "vacuum" state have the samesymmetries of Z(s), then a and b must be constrained to obey 2b + a =1 ...

From the ... inner product of two arbitrary states , by choosingfor example that l = -2 => k = 1/4 , one concludes that thepseudo-norm

and consequently case A ... is ruled out and case B ... stands ...since the pseudo-norm ... is not null this implies that theeigenvalues E_s,E*_s ... are real-valued E_s = s(1-s) = E*_s =s*(1-s*) which means that the Riemann Hypothesis is true.

The results ... and conclusions remain the same for other choicesof the parameters l, k so far as l, k are constrained to obey thecondition l = 4(2k - 1) <=> a + 2b = 1 imposed from thesymmetry considerations since the orthogonal states PSI_s_n(t) to thereference state PSI_s_o (t) must obey the same symmetry conditionswith respect to the critical line and real line as the non-trivialzeta zeros ... as a result of l = 4(2k -1).

The key reason why the Riemann hypohesis is true is due to thefact that there is no zero at s_o = 1/2 + i 0 and consequently thepseudo-norm < PSI_s | CT | PSI_s > is not null. Had there beena zero at the center of symmetry s_o = 1/2 + i 0 the RH would havebeen false. ...".

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