Arithmetical Investigations: Representation Theory, by Shai M. J. Haran

By Shai M. J. Haran

In this quantity the writer extra develops his philosophy of quantum interpolation among the genuine numbers and the p-adic numbers. The p-adic numbers comprise the p-adic integers Zpwhich are the inverse restrict of the finite earrings Z/pn. this provides upward push to a tree, and chance measures w on Zp correspond to Markov chains in this tree. From the tree constitution one obtains designated foundation for the Hilbert area L2(Zp,w). the genuine analogue of the p-adic integers is the period [-1,1], and a likelihood degree w on it supplies upward thrust to a unique foundation for L2([-1,1],w) - the orthogonal polynomials, and to a Markov chain on "finite approximations" of [-1,1]. For distinct (gamma and beta) measures there's a "quantum" or "q-analogue" Markov chain, and a unique foundation, that inside of sure limits yield the true and the p-adic theories. this concept should be generalized variously. In illustration concept, it's the quantum basic linear staff GLn(q)that interpolates among the p-adic workforce GLn(Zp), and among its genuine (and complicated) analogue -the orthogonal On (and unitary Un )groups. there's a related quantum interpolation among the true and p-adic Fourier rework and among the genuine and p-adic (local unramified a part of) Tate thesis, and Weil particular sums.

Show description

Read Online or Download Arithmetical Investigations: Representation Theory, Orthogonal Polynomials, and Quantum Interpolations PDF

Similar quantum physics books

Wholeness and the Implicate Order (Routledge Classics)

Submit yr observe: First released July 1st 1980

David Bohm used to be one of many most effective medical thinkers and philosophers of our time. even if deeply motivated by means of Einstein, he used to be additionally, extra strangely for a scientist, encouraged by way of mysticism.

Indeed, within the Seventies and Nineteen Eighties he made touch with either J. Krishnamurti and the Dalai Lama whose teachings assisted in shaping his paintings. In either technological know-how and philosophy, Bohm's major situation was once with knowing the character of fact typically and of attention particularly.

In this vintage paintings he develops a idea of quantum physics which treats the totality of lifestyles as an unbroken entire. Writing basically and with out technical jargon, he makes complicated principles obtainable to someone attracted to the character of truth.

Lectures of Quantum Field Theory

This e-book contains the lectures for a two-semester direction on quantum box concept, and as such is gifted in a particularly casual and private demeanour. The path starts off with relativistic one-particle structures, and develops the fundamentals of quantum box idea with an research of the representations of the Poincaré staff.

Adiabatic Perturbation Theory in Quantum Dynamics

Separation of scales performs a primary position within the figuring out of the dynamical behaviour of advanced platforms in physics and different ordinary sciences. A in demand instance is the Born-Oppenheimer approximation in molecular dynamics. This e-book makes a speciality of a contemporary method of adiabatic perturbation thought, which emphasizes the function of potent equations of movement and the separation of the adiabatic restrict from the semiclassical restrict.

Extra info for Arithmetical Investigations: Representation Theory, Orthogonal Polynomials, and Quantum Interpolations

Example text

3) of Martin kernel, the sequences K((1, 0), (in , jn )) = = (in + jn − 1)! jn ! jn ! (in + jn )! ζη (α + 2, β) in ζη (α, β) in + jn ζη (α + 2, β) and K((0, 1), (in , jn )) = jn ζη (α, β) . in + jn ζη (α, β + 2) should be Cauchy sequences. This means that either 1. There exists some i∞ and j∞ such that (in , jn ) ≡ (i∞ , j∞ ) for all n 0, or 2. in + jn → ∞ (n → ∞) and jn /in converges (in wide sense) in [0, ∞]. Conversely, as we will check below soon, the Martin kernel converges for all (i, j) whenever jn /in converges in the wide sense.

The name of the γ- measure is given from the gamma function. If p = η, τZβη is the Gaussian probability measure, which is invariant under Z∗η = {±1}. If p = η, then τZβp is also invariant under the action Z∗p , whence gives a probability measure on Zp /Z∗p . Notice that for all n ≥ 0, we have τZβp (pn Z∗p ) = p−nβ (1 − p−β ) (see Fig. 7). 1 The Projective Space P1 (Qp ) We next define a measure on the projective line P1 (Qp ), which is called β-measure. Let V (Qp ) := Qp × Qp be the plane and V ∗ (Qp ) := {(x, y) ∈ V (Qp ) | (x, y) = (0, 0)}.

This is a simple calculation. 3) again, we have K((i, j), (in , jn )) = (in + jn − i − j)! jn ! ζη (α, β) (in + jn )! (jn − j)! ζη (α + 2i, β + 2j) = in (in − 1) · · · (in − i + 1) · jn (jn − 1) · · · (jn − j + 1) ζη (α, β) (in + jn )(in + jn − 1) · · · (in + jn − i − j − 1) ζη (α + 2i, β + 2j) 1 1+ = 1 jn in 1+ jn +1 in −1 1· 1− ··· 1 1+ 1 in + jn jn +i−1 in −i+1 ··· 1 − 1 · 1+ 1 in jn 1+ in +1 jn −1 ··· i + j − 1 ζη (α + 2i, β + 2j) in + jn ζη (α, β) 1 1+ in +j−1 jn −j+1 . Therefore we have lim in +jn →∞ jn /in →|x|2 η K((i, j), (in , jn )) = 1 1 + |x|2η i 1 1 + |x|−2 η j ζη (α, β) .

Download PDF sample

Rated 4.17 of 5 – based on 34 votes