共查询到20条相似文献,搜索用时 15 毫秒
1.
Kai Johannes Keller Nikolaos A. Papadopoulos Andrés F. Reyes-Lega 《Mathematische Semesterberichte》2008,55(2):149-160
The aim of this paper is to give a simple, geometric proof of Wigner’s theorem on the realization
of symmetries in quantum mechanics that clarifies its relation to projective geometry. Although several
proofs exist already, it seems that the relevance of Wigner’s theorem is not fully appreciated in general.
It is Wigner’s theorem which allows the use of linear realizations of symmetries and therefore guarantees
that, in the end, quantum theory stays a linear theory. In the present paper, we take a strictly geometrical
point of view in order to prove this theorem. It becomes apparent that Wigner’s theorem is nothing else
but a corollary of the fundamental theorem of projective geometry. In this sense, the proof presented here
is simple, transparent and therefore accessible even to elementary treatments in quantum mechanics. 相似文献
2.
Ordering in mechanical geometry theorem proving 总被引:2,自引:0,他引:2
Hongbo Li 《中国科学A辑(英文版)》1997,40(3):225-233
Ordering in mechanical geometry theorem proving is studied from geometric viewpoint and some new ideas are proposed. For Thebault’s
theorem which is the most difficult theorem that has ever been proved by Wu’s method, a very simple proof using Wu’s method
under a linear order is discovered.
Project supported by the National Natural Science Foundation of China. 相似文献
3.
In this article we present a simple and elegant algebraic proof of Pascal’s hexagon theorem which requires only knowledge
of basics on conic sections without theory of projective transformations. Also, we provide an efficient algorithm for finding
an equation of the conic containing five given points and a criterion for verification whether a set of points is a subset
of the conic. 相似文献
4.
We present a geometrical version of Herbert’s theorem determining the homology classes represented by the multiple point manifolds
of a self-transverse immersion. Herbert’s theorem and generalizations can readily be read off from this result. The simple
geometrical proof is based on ideas in Herbert’s paper. We also describe the relationship between this theorem and the homotopy
theory of Thom spaces. 相似文献
5.
A simple but rigorous proof of the Fundamental Theorem of Calculus is given in geometric calculus, after the basis for this theory in geometric algebra has been laid out. Various classical
examples of this theorem, such as the Green’s and Stokes’ theorem are discussed, as well as the theory of monogenic functions
which generalizes analytic functions of a complex variable to higher dimensions. 相似文献
6.
Some ideas of T. Kamae’s proof using nonstandard analysis are employed to give a simple proof of Birkhoff’s theorem in a classical
setting as well as Kingman’s subadditive ergodic theorem. 相似文献
7.
The aim of this paper is to give an alternative proof of Kac’s theorem for weighted projective lines over the complex field.
The geometric realization of complex Lie algebras arising from derived categories is essentially used. 相似文献
8.
Francesco Baldassarri 《Milan Journal of Mathematics》2005,73(1):237-258
We review the notion of regular singular point of a linear differential equation with meromorphic coefficients, from the viewpoint
of algebraic geometry. We give several equivalent definitions of regularity along a divisor for a meromorphic connection on
a complex algebraic manifold and discuss the global birational theory of fuchsian differential modules over a field of algebraic
functions. We describe the generalized algebraic version of Deligne’s canonical extension, constructed in [1, I.4].
Our main interest lies in the algebraic form of Deligne’s regularity criterion [2, II.4.4 (iii)], asserting that, on a normal
compactification, only one codimensional components of the locus at infinity need to be considered. If one considers the purely
algebraic nature of the statement, it is surprising that the only existing proof of this criterion is the transcendental argument
given by Deligne in his corrigendum to loc. cit. dated April 1971. The algebraic proof given in our book [1, I.5.4] is also incorrect, as J. Bernstein kindly indicated to
us.We introduce some notions of logarithmic geometry to let the reader appreciate Bernstein’s (counter)examples to some statements
in our book [1]. Standard methods of generic projection in projective spaces reduce the question to a two-dimensional puzzle.
We report on ongoing correspondence with Y. André and N. Tsuzuki, leading to partial results and provide examples indicating
the subtlety of the problem.
Lecture held in the Seminario Matematico e Fisico on January 31, 2005 Received: June 2005 相似文献
9.
McMullen’s proof of the Hard Lefschetz Theorem for simple polytopes is studied, and a new proof of this theorem that uses
conewise polynomial functions on a simplicial fan is provided. 相似文献
10.
William F. Donoghue 《Israel Journal of Mathematics》1966,4(3):153-170
The study of the exact interpolation of quadratic norms in vector spaces depends in an essential way on the theory of monotone
matrix functions developed by Loewner in 1934 [4]. This theory, in its turn, depends on Loewner’s solution of a problem of
interpolation by rational functions of a certain class. The discussion of this latter problem is necessarily complicated,
and Loewner’s text does not lend itself to ready reference. It has therefore seemed worthwhile to recast a portion of Loewner’s
results in a form more suited to the applications we have in view. Our work, however, is not wholly derivative; none of our
theorems are explicitly stated by Loewner and our arguments, which are of a more geometric character, are essentially different.
The knowledgeable reader will note that our hypotheses are slightly stronger than Loewner’s and that our results are therefore
also stronger. For the applications which we have in mind, Theorem III is the most important result; the proof of this theorem
depends on all of the previously developed theory. 相似文献
11.
V. I. Arnold 《Proceedings of the Steklov Institute of Mathematics》2011,273(1):25-34
Courant proved that the zeros of the nth eigenfunction of the Laplace operator on a compact manifold M divide this manifold into at most n parts. He conjectured that a similar statement is also valid for any linear combination of the first n eigenfunctions. However, later it was found out that some corollaries to this generalized statement contradict the results
of quantum field theory. Later, explicit counterexamples were constructed by O. Viro. Nevertheless, the one-dimensional version
of Courant’s theorem is apparently valid; to prove it, I.M. Gel’fand proposed a method based on the ideas of quantum mechanics
and the analysis of the actions of permutation groups. This leads to interesting questions of describing the statistical properties
of group representations that arise from their action on eigenfunctions of the Laplace operator. The analysis of these questions
entails, among other things, problems of singularity theory. 相似文献
12.
Cyclic orders of graphs and their equivalence have been promoted by Bessy and Thomassé’s recent proof of Gallai’s conjecture.
We explore this notion further: we prove that two cyclic orders are equivalent if and only if the winding number of every
circuit is the same in the two. The proof is short and provides a good characterization and a polynomial algorithm for deciding
whether two orders are equivalent.
We then derive short proofs of Gallai’s conjecture and a theorem “polar to” the main result of Bessy and Thomassé, using the
duality theorem of linear programming, total unimodularity, and the new result on the equivalence of cyclic orders. 相似文献
13.
S. Raghavan 《Proceedings Mathematical Sciences》1984,93(2-3):147-160
Recently, Bombieri and Vaaler obtained an interesting adelic formulation of the first and the second theorems of Minkowski
in the Geometry of Numbers and derived an effective formulation of the well-known “Siegel’s lemma” on the size of integral
solutions of linear equations. In a similar context involving linearinequalities, this paper is concerned with an analogue of a theorem of Khintchine on integral solutions for inequalities arising from
systems of linear forms and also with an analogue of a Kronecker-type theorem with regard to euclidean frames of integral
vectors. The proof of the former theorem invokes Bombieri-Vaaler’s adelic formulation of Minkowski’s theorem. 相似文献
14.
A. G. Khovanskii 《Functional Analysis and Its Applications》2011,45(4):305-315
Birationally invariant intersection theory is a far-reaching generalization and extension of the Bernstein-Kushnirenko theorem.
This paper presents transparent proofs of Hilbert’s theorem on the degree of a projective variety and other related statements
playing an important role in this theory. The paper is completely self-contained; we recall all necessary definitions and
statements. 相似文献
15.
The Wigner's theorem, which is one of the cornerstones of the mathematical formulation of quantum mechanics, asserts that every symmetry of quantum system is unitary or anti-unitary. This classical result was first given by Wigner in 1931. Thereafter it has been proved and generalized in various ways by many authors. Recently, G.P. Gehér extended Wigner's and Molnár's theorems and characterized the transformations on the Grassmann space of all rank-n projections which preserve the transition probability. The aim of this paper is to provide a new approach to describe the general form of the transition probability preserving (not necessarily bijective) maps between Grassmann spaces. As a byproduct, we are able to generalize the results of Molnár and G.P. Gehér. 相似文献
16.
Krishnaswami Alladi 《The Ramanujan Journal》2009,20(3):253-256
We will interpret a partial theta identity in Ramanujan’s Lost Notebook as a weighted partition theorem involving partitions
into distinct parts with smallest part odd. A special case of this yields a new result on the parity of the number of parts
in such partitions, comparable to Euler’s pentagonal numbers theorem. We will provide simple and novel proofs of the weighted
partition theorem and the special case. Our proof leads to a companion to Ramanujan’s partial theta identity which we will
explain combinatorially. 相似文献
17.
In a recent paper, the authors have proved results characterizing convexity-preserving maps defined on a subset of a not-necessarily
finite dimensional real vector space as projective maps. The purpose of this note is three-fold. First, we state a theorem
characterizing continuous, injective, convexity-preserving maps from a relatively open, connected subset of an affine subspace
of ℝ
m
into ℝ
n
as projective maps. This result follows from the more general results stated and proved in a coordinate-free manner in the
above paper, and is intended to be more accessible to researchers interested in optimization algorithms. Second, based on
that characterization theorem, we offer a characterization theorem for collinear scalings first introduced by Davidon in 1977
for deriving certain algorithms for nonlinear optimization, and a characterization theorem for projective transformations
used by Karmarkar in 1984 in his linear programming algorithm. These latter two theorems indicate that Davidon’s collinear
scalings and Karmarkar’s projective transformations are the only continuous, injective, convexity-preserving maps possessing
certain features that Davidon and Karmarkar respectively desired in the derivation of their algorithms. The proofs of these
latter two theorems utilize our characterization of continuous, injective, convexity-preserving maps in a way that has implications
to the choice of scalings and transformations in the derivation of optimization algorithms in general. The third purpose of
this note is to point this out.
Received: January 2000 / Accepted: November 2000?Published online January 17, 2001 相似文献
18.
The main conclusion of this paper is that the Bell–Wigner–Accardi theory of quantum probabilities in spin systems may be placed within the general operator trigonometry developed independently by this author about 30 years ago. The use of the Grammian from the operator trigonometry simplifies and clarifies the analysis of Wigner. A general triangle inequality from the operator trigonometry clarifies and generalizes the analysis of Accardi. The statistical meaning of the complex numbers in quantum mechanics is seen to be that of the natural geometry of the operator trigonometry. A new connection of the operator trigonometry to CP symmetry violation is established. 相似文献
19.
20.
Motivated by the central limit problem for convex bodies, we study normal approximation of linear functionals of high-dimensional
random vectors with various types of symmetries. In particular, we obtain results for distributions which are coordinatewise
symmetric, uniform in a regular simplex, or spherically symmetric. Our proofs are based on Stein’s method of exchangeable
pairs; as far as we know, this approach has not previously been used in convex geometry. The spherically symmetric case is
treated by a variation of Stein’s method which is adapted for continuous symmetries.
This work was done while at Stanford University. 相似文献