In an e-mail discussion of his February 2007 paper "The Riemann Hypothesis 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 by F. Reese Harvey "Spinors and Calibrations" (Academic Press 1990) at pages 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 why CT-invariant QM forces the spectrum s ( 1- s ) of H_A

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

**In his February 2007 paper, The Riemann Hypothesis is a
consequence of CT -invariant Quantum Mechanics, Carlos Castro
Perelman said: **

"...The Riemann's hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form s_n = 1/2 +/- i lambda_n. By constructing a continuous family of scaling-like operators involving the Gauss-Jacobi theta series and by invoking a novel CT -invariant Quantum Mechanics, involving a judicious charge conjugation C and time reversal T operation, we show why the Riemann Hypothesis is true. ...

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

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

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

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

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

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

that is related to D_1 by the substitution t -> 1/t and by noticing that dV(1/t)/dln(1/t) = -dV(1/t)/dlnt where V(1/t) is not equal 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 under the x -> 1/x transformations. Since V(t) can be chosen arbitrarily, we choose it to be related to the Bernoulli string spectral counting function, given by the Jacobi theta series

where w(x) = SUM{n=1, oo} e^(- pi n^2 x ) . The Gauss-Jacobi series 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 the same spectrum s(1-s) which is real-valued only in the critical line and in the real line. ...[the above two equations are]... the one-dimensional version of the eigenfunctions of the two-dimensional hyperbolic Laplacian given in terms of the Eisenstein's series. Had H_A,H_B been Hermitian one would have had an immediate proof of the RH. ...

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 the vanishing commutators [H_A, CT ] = [H_B, CT ] = 0 ... When the operators H_A, H_B commute with CT , there exits new eigenfunctions PSI^CT_s (t) of the HA operator with eigenvalues s* (1 - s* ). ... similar results follow for the H_B operator ... one has two 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 ... the essence of the proof relies in establishing a one to one correspondence among the zeta zeros s_n with the states PSI_s_n (t) orthogonal to the ground ( vacuum state ) PSI_s_o (t) associated with the center of symmetry s_o = 1/2 + i 0 of the non-trivial zeta zeros and 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_n since Z[s] is proportional to zeta(s) ...

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

The crucial problem is whether there are zeros outside the critical line (but still inside the critical strip) and not the interpretation ... as a genuine inner product. Despite this, we still rather loosely refer to this mapping as a scalar product. The states still have a real norm squared, which however need not to be positive-definite. Here we must emphasize that our arguments do not rely on the validity of the zeta-function regularization procedure ... which precludes a rigorous interpretation ... as a scalar product. Instead, we can simply replace the expression "scalar product of PSI_s_1 and PSI_s_2" by the map S of complex numbers defined 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 other words, our arguments do not rely on an evaluation of the integral ... but only on the mapping ... defined as the finite part of the integral ... 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 mean always the symmetries of the kernel of the S map.

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

if a and b are such that 2b + a = 1, then the symmetries of all the states s orthogonal to the "vacuum" state are preserved by any map S ... which leads to Z(as + b). In fact, if the state associated with 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 the Riemann zeta-function has zeros at s' = x' + iy', s'* , 1-s' and 1-s'* . If we equate as + b = s', then as* + b = s'* . Now, 1-s' will be 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_s orthogonal to the "vacuum" state, parameterized by the complex number 1/2 + i0, will then have the same symmetry properties with respect to the critical line as the nontrivial zeros of zeta. ...

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

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

and consequently case A ... is ruled out and case B ... stands ... since the pseudo-norm ... is not null this implies that the eigenvalues 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 choices of the parameters l, k so far as l, k are constrained to obey the condition l = 4(2k - 1) <=> a + 2b = 1 imposed from the symmetry considerations since the orthogonal states PSI_s_n(t) to the reference state PSI_s_o (t) must obey the same symmetry conditions with respect to the critical line and real line as the non-trivial zeta zeros ... as a result of l = 4(2k -1).

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

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