重复发表也没办法。最近关于猜想的博文都是应要而作。这是应张志东的要求，也是附和dongping2009的提议，“应该同时贴出Comment与Rejoinder之间的Response，这样似乎规范一些。” 请注意，对张志东和伍法岳的电邮要求，我只是复制转贴，不负责更多排版工作。 v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} w\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} Normal 0 false false false EN-US X-NONE X-NONE MicrosoftInternetExplorer4 st1\:*{behavior:url(#ieooui) } /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-qformat:yes; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-ascii-font-family:Calibri; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:minor-bidi;} Philosophical Magazine Vol. 88, No. 26, 11 September 2008, 3097–3101 Response to ‘Comment on a recent conjectured solution of the three-dimensional Ising model’ Z.D. Zhang* Shenyang National Laboratory for Materials Science, Institute of Metal Research and International Centre for Materials Physics, Chinese Academy of Sciences, Shenyang, China (Received 20 August 2008; final version received 3 October 2008) This is a Response to a recent Comment on the conjectured solution of the three-dimensional (3D) Ising model . Several points are made: (1) Conjecture 1, regarding the additional rotation, is understood as performing a transformation for smoothing all the crossings of the knots. (2) The weight factors in Conjecture 2 are interpreted as a novel topologic phase. (3) The conjectured solution and its low- and high-temperature expansions are supported by the mathematical theorems for the analytical behavior of the Ising model. The physics behind the extra dimension is also discussed briefly. In the preceding paper, Wu et al. comment on the conjectured solution of the three dimensional (3D) Ising model presented in . The comments that Wu et al. make regarding the presentation (length, usage of some words, placement in Appendix) will not be replied to here. Their comments concerning content concentrate on the low- and high temperature expansions given in and on the different choices of the weight functions wy and wz. The latter problem needs clarification; the first two objections have been anticipated and are rejected in . Although it is not necessary to repeat here what is already in the original paper, I shall underline several issues with new ideas. First of all, as is clear from the references quoted in , I do not contest the statement that the Ising model has been well-studied for over 80 years, mainly in great contributions of many distinguished scientists, including the authors of . However, present knowledge cannot serve as a standard for judging the conjectured solution, because the 3D case is not yet fully understood. There are two ‘dark clouds’: (1) The divergence of the so-called ‘exact’ low-temperature expansions and the existence of an unphysical singularity. (2) The possibility of the occurrence of a phase transition at infinite temperature ( T = , = 1/( k B T ) = 0) according to the Yang-Lee theorems . It is regrettable that the objections of the authors of are limited to the outcome of the calculations and that they did not comment on the topology-based approach underlying the derivation. The putative solution was deduced using (among other steps) two conjectures, which at the moment cannot be qualified as rigorous. Therefore, the validity of the solution hinges on the validity of the conjectures. The logic of Conjecture 1 is very simple: the topologic problem of the 3D Ising system, which is the origin of the difficulties, can be dealt with by introducing a boundary condition; (i.e. an additional rotation matrix V' 4 ) to smooth the crossings of numerous knots hidden in the boundary condition (Equation (15)) for the matrix V V 3 V 2 V 1 . (The equation number in the preceding sentence and those given later in this article refer to equations in ). There are two choices for smoothing a given crossing (x), and thus 2 N states of a diagram with N crossings . Mathematically, the state summation , producing the bracket polynomial, appears as a generalized partition function, defined on the knot diagram, and provides a connection between knot theory and physics . Here, K S is the product of vertex weights, ç – the number of loops in the state S. Therefore, the matrix V consists of two kinds of contributions: those reflecting the local arrangement of spins and others reflecting the non-local behavior of the knots. After smoothing, there will be no crossing in the new matrix V' V' 4 V' 3 V' 2 V' 1 , which precisely includes the topologic contribution to the partition function, which becomes diagonalizable. The intrinsic nonlocal behavior caused by the knots, requires by itself the additional rotation matrix as well as the extra dimension to handle the procedure in the much larger Hilbert space, since in 3D the operators of interest generate a much larger Lie algebra . This merely performs a transformation on the Hamiltonian and the wavevectors of the system. Because the wellrecognized ‘correct’ high- and low-temperature expansions never take into account the global topologic effect, they cannot be correct at finite temperatures in 3D. The only exception is that the high-temperature expansions in 3D can be correct at/near = 0, where the interaction does not exist (or is extremely weak) so that the global effect is negligible. I recognize that one of the key assumptions, Conjecture 2, concerning the weight factors w x , w y and w z , was not presented in a logical sequence in , mainly because the details were moved to the Appendices in view of length considerations. The weight factors were defined in the range and, considering symmetry, their roles can be interchanged without altering the eigenvalues (Equation (29)) or the partition function (Equation (49)) (see p.5372). It is possible to generalize the weight factors in the eigenvectors (Equation (33)) as complex numbers w x ， w y , and w z with phases , and . However, only the real part of the phase factors appears in the eigenvalues (29), (30), (31), (49), etc. of the system, so that w x , w y and w z can be replaced by w x Re ， w y Re and w z Re , respectively. They may be understood as the results of performing a transformation of the eigenvectors of the 3D Ising system to the ‘quaternion’ Hilbert space and, subsequently, projecting them back to 3D . Various geometrical phase factors, such as the Aharonov-Bohm phase or Berry phase, among others , have been discovered in the past decades, which are related to the global topologic behavior of quantum systems. The potential in quantum mechanics was viewed as a connection that relates to phases at different locations , which should also be true for the 3D Ising interactions. The present phase factor, which originates from the geometrical behavior of the 3D Ising system, is novel. This topologic phase is a function of the interactions and temperature, depending sensitively on whether the knots exist or not. Thus, the value of the weight factors changes at/near T = owing to the change of the geometrical (topologic) structure, while it crosses over from w x Re = 1, w y Re = 0 and w z Re = 0 (their role can interchange, as mentioned, to maintain the four-fold integral) for 3D to ê w x ê º 1, ê w y ê º 0 and ê w z ê º 0 (to reduce to the two-fold integral) for 2D. The latter results in a crossover of the critical exponents. The phase factor is akin to the one appearing in Feynman’s path-integral theory , where the transition amplitude between an initial and a final state is the sum over all paths, connecting two points, of the weight factor , with S the action of the system. Our action here is topologic, which arises from the overall geometry of the path , similar to other topologic phases. One of the criticisms repeatedly voiced in is based on the ‘fact’ that the convergence of the low- and high-temperature series was rigorously proved. It is argued that the expressions (Equations (49), (74) and (99)) cannot be the true solution because the weight factors result in a difference between expressions for the high-temperature limit (Equation (A.12)) and the result for more general temperature (Equation (A.13)), for which w x = 1, w y = 0 and w z = 0, was chosen. But attention has never been paid to the possibility of the existence of a phase transition at/near = 0 ( , page 5371). The Lee-Yang theorems , which are rigorous and very general, can be suitable for the 3D Ising model. It would not violate other rigorous results if the singular behavior at = 0 served as a necessary condition adding to the convergence of the series. Proving only the radius of convergence of the series is insufficient (especially in 3D). Lebowitz and Penrose proved a theorem for the high-temperature series and distinguished 0 and = 0. They stated clearly that, since = 0 lies on the boundary of the region E of ( , z) space, there is no general reason to expect a series expansion of p or n in powers of to converge (p.102 of ). A qualitative picture is given in Griffiths’ review , showing the shape of the region in the T–H plane (Figure 6) where all is analytic, but he started with the condition 0. The inequality (Equation (2B.8)) (or other similar ones) of , which is important for proving rigorous results, is valid only for 0. Actually, if we plotted Griffiths’ T–H plane as a –H plane, there should be a singularity at = 0. Therefore, distinguishing ‘at/near infinity’ and ‘finite temperature’ is reasonable. Sachdev claimed in Figures 4.3 and 11.2 of his book that the so-called ‘lattice high-T ’ phase at very high temperatures has non-universal critical behavior. Though the singularity in the 1D quantum model (mapping to the 2D Ising model) might not be strong enough to give any sort of transition, it is our understanding that the geometrical change in the 2D quantum model (mapping to the 3D Ising model) may introduce a transition at T = . Usually, mathematical theorems prove analytical behaviors in a very general form of functions based on some assumptions (for instance, the Peierls condition, 0 for Theorem 2.1. in Sinai ; sufficiently small or in Theorem 18.1.2, Corollary 18.1.4, Theorem 18.3.1, Proposition 18.4.2, Theorem 18.5.1, and assumptions P1, P2 and E c+5 in Theorems 20.3.1-2, 20.4.1-2 and small in (20.5.4) in Glimm and Jaffe ), which do not guarantee the analytic behavior of the low and high-temperature expansions in their well-known expansion basis, (for example, the divergence of the low-temperature series is contradictory to these theorems). From another angle, we could think that the analytic nature of the expansions for the conjectured solution is supported by these mathematical theorems . In addition, the conjectured solution reduces to Zandvliet et al.’s results of the anisotropic 3D Ising model where two of the three exchange energies are small compared to the third one , which agree with Fisher’s rigorous formulae in this limit . The necessity of introducing the extra dimension can be understood from another angle. The basic issues are that some key points are often overlooked in quantum statistical mechanics. To introduce the concept of thermal equilibrium (strictly speaking, an undefined (or multidefined) concept), our Ising model is made part of a system big enough for statistical concepts to be useful . In a quantum statistics system, besides the average in a quantum state (expectation value), one also averages with respect to the probability distribution of systems in an ensemble ; i.e. a whole collection (a large number N) of identical Ising models of m rows and n columns and l planes connected together by infinitely weak forces, which allow the Ising models to exchange energy but that do not contribute to the total energy of the system. Namely, a piece of substance is isolated from everything; any part of the substance must be in equilibrium with the rest, serving as a heat reservoir that well defines a temperature . But the temperature in statistical mechanics is actually the time in quantum field theory , since the Euclidean time interval can be consistently identified with . The partition function Z = Tr can be represented in the Schrödinger picture as , which is merely the transition amplitude with the identification t = - i . This indicates that the time t is hidden in the framework of the statistical mechanics for an equilibrium system. Therefore, one has a clue that the framework of the statistical mechanics for the 3D Ising systems should include the time, being in the (3+1) dimensional Euclidean spacetime. The same should be true for the 2D quantum model as is shown by the well-known mapping . In quantum mechanics, at any instance of time, the wave function of a truly isolated system can be expressed by a linear superposition of a complete orthonormal set of stationary wave functions n : , where c n is a complex number and is generally a function of time . In quantum statistical mechanics, the wave function depends on both the coordinates of the system under consideration and the coordinates of the external world (an additional dimension is indeed needed). n denotes a complete set of orthonormal stationary wave functions of the system, while cn is interpreted as a wave function of the external world (depending on its coordinates). Thus, the scalar product (c n , c m ) of the nth and the mth wave function of the external world is also a function of time. This means that the average value of a large number of measurements of an operator, instantaneously given its expectation value, depends indeed on the time, although in the laboratory we measure not its instantaneous value but a time average . However, with the postulates of equal a priori probability and random phase, the wave function of the system can be regarded as with the phases of the complex numbers bn being random, to take into account the effect of the external world in an average way. It was emphasized that for this reduction to be effectively valid, the system must interact with the external world. Otherwise, the postulate of random phase is false, because the randomness of the phases means no more and no less than the absence of interference of probability amplitude. However, such a circumstance cannot be true for all time though it may be true at an instant . The postulates of quantum statistical mechanics are regarded as working hypotheses whose justification lies in their agreement with experiments . Such a point of view is not entirely satisfactory and a rigorous derivation is lacking (see pp.184–188 of ). So, the immediate questions are how the system interacts with the external world (it may be somehow inconsistent with what we accepted for infinitely weak forces), and what the missing part is whilst employing the postulates. To answer these questions in detail is beyond the scope of this reply, but the discussions above show the necessity of the extra dimension, and also imply the existence of flaws in the Comment. In summary, admitting that there are some open questions related to the choice of the weight factors, which will need more research, we have argued that the correct reproduction of the high-temperature expansion cannot be a coincidence and the failure in reproducing term by term the low-temperature expansion does not disqualify the new approach to deal with knots by means of an extension into a fourth dimension. Acknowledgement The author appreciates the support of the National Natural Science Foundation of China (under grant numbers 50831006, 10674139 and 50331030). References F.Y. Wu, B.M. McCoy, M.E. Fisher et al., Phil. Mag. 88 (2008) p.3093. Z.D. Zhang, Phil. Mag. 87 (2007) p.5309. C.N. Yang and T.D. Lee, Phys. Rev. 87 (1952) p.404; T.D. Lee and C.N. Yang, Phys. Rev. 87 (1952) p.410. L.H. Kauffman, Rep. Prog. Phys. 68 (2005) p.2829. G.F. Newell and E.W. Montroll, Rev. Mod. Phys. 25 (1953) p.353. Y. Aharonov and D. Bohm, Phys. Rev. 115, (1959) p.485; M.V. Berry, Proc. R, Soc. A 392 (1984) p.45; T.W. Barrett, Topological Foundations of Electromagnetism, World Scientific, Singapore, 2008. A. Das, Field Theory: A Path Integral Approach, World Scientific, Singapore, 1993. Ya.G. Sinai, Theory of Phase Transitions: Rigorous Results, Pergamon Press, Oxford, 1982; Chapter II; J. Glimm and A. Jaffe, Quantum Physics, 2nd ed., Springer, New York, 1987; Chapter 18, 20; R.B. Israel, Commun. Math. Phys. 50 (1976) p.245; M. Zahradnik, J. Stat. Phys. 47 (1987) p.725. J.L. Lebowitz and O. Penrose, Commun. Math. Phys. 11 (1968) p.99. R.B. Griffiths, Rigorous results and theorems, in Phase Transitions and Critical Phenomena, Vol. 1, C. Domb and M.S. Green eds., Academic Press, New York, 1972, p.7. S. Sachdev, Quantum Phase Transitions, Cambridge University Press, Cambridge, UK, 1999. H.J.W. Zandvliet, A. Saed and C. Hoede, Phase Transitions 80 (2007) p.981. C.-Y. Weng, R.B. Griffiths and M.E. Fisher, Phys. Rev. 162 (1967) p.475; M.E. Fisher, Phys. Rev. 162 (1967) p.480. B.M. McCoy and T.T. Wu, The Two-Dimensional Ising Model, Harvard University Press, Cambridge, MA, 1973. K. Huang, Statistical Mechanics, Wiley, New York, 1963.
伍法岳先生在发给我的公开电邮里是有PDF附件的，comment和rejoinder。我尝试了几次，也没能上传到博客上来。今天，伍先生又把附件排出我可以复制剪贴的方式发给我，现在我再发到博客上来，供真正对3D猜想在学术上感兴趣的参考。 Philosophical Magazine Vol. 88, No. 26, 11 September 2008, 3093–3095 Comment on a recent conjectured solution of the three-dimensional Ising model F.Y. Wu a , B.M. McCoy b , M.E. Fisher c * and L. Chayes d a Department of Physics, Northeastern University, Boston, USA; b C.N. Yang Institute for Theoretical Physics, State University of New York, New York, USA; c Institute for Physical Science and Technology, University of Maryland, MD, USA; d Department of Mathematics University of California, Los Angeles, USA ( Received 9 July 2008; final version received 3 October 2008 ) In a recent paper published in Philosophical Magazine , the author advances a conjectured solution for various properties of the three-dimensional Ising model. Here, we disprove the conjecture and point out the flaws in the arguments leading to the conjectured expressions. Keywords : 3D Ising model; exact solution; conjectured results The Ising model is a well-known and well-studied model of magnetism. Owing to its apparent simplicity, the model has attracted the concerted attention of physicists for over 80 years. Ising solved the model in one dimension in 1925. In 1944, Onsager obtained the exact free energy of the two-dimensional (2D) model in zero field and, in 1952, Yang presented a computation of the spontaneous magnetization. But, the three-dimensional (3D) model has withstood challenges and remains, to this date, an outstanding unsolved problem. In a recent 111-page paper published in Philosophical Magazine, Zhang advanced conjectured expressions for the free energy and spontaneous magnetization of the 3D model. Here, we show that the conjectures are false. Zhang considered the nearest-neighbor 3D Ising model on the simple cubic lattice: see his Equations (1) and (2) and associated text where the notations are established with, specifically, three coupling constants K =J/ k B T , K ’ , and K ’’ : Arguments leading to the conjectured solutions are roughly as follows y : The author presents an expression, Equation (49), in the form of a four-fold integral (which reduces to equation (74) in the isotropic case) as the exact free energy. But this expression contains yet-to-be-determined, unknown weight functions w x , w y , w z : The argument next jumps to Appendix A, where the author sets w x = 1 and expands w y = w z in the form of a square root of a series: see Equation (A2). Also in the Appendix, the author demonstrates that the expansion coefficients of the first 11 terms of the series ______________________________________________________________ *Corresponding author. Email: xpectnil@umd.edu y Key assumptions made in are not presented in a logical sequence but are often hidden in inconspicuous places, making it difficult for a reader to see what is really going on. 3094 F.Y. Wu et al. can be fitted, as shown in (A2), to ensure that Equation (74) reproduces the known 11 terms of the exact high-temperature expansion of the free energy obtained by Guttmann and Enting ; see also line 1, p.5326. Almost as if in ‘fine print’, the author then sets w y = w z = 0 (see line 7 on p.5326 just before Equation (50)) and uses the resulting form of (49) as the conjectured solution of the free energy throughout the ensuing analysis where conclusions on the critical point, etc. are drawn. The reason given for taking w y = w z = 0 is what the author calls ‘‘ Ansatz 1 ’’ in Appendix A (p.5399). Under this ansatz, the author argues (lines 7–9, p.5400), the series inside the square root would become negative making w y and w z imaginary. Since imaginary quantities are ‘‘ physically not meaningful ’’, w y and w z ‘are always equal to zero’ (p.5400). It must be emphasized that this argument for choosing the weights w y = w z = 0 is deeply flawed. Indeed, in light of the fitting of the series in Equation (A2) to reproduce the known high-temperature expansions, one knows that the choice w y = w z = 0 will not reproduce the exact high-temperature expansions. Hence, the resulting expressions (49) and (74) cannot be the true solution of the free energy. By the same token, the ‘‘ putatively determined ’’ critical point relations (see the Abstract, etc.) carry no credence. For the spontaneous magnetization, the author presents the expression (99) (reducing to Equation (102) in the isotropic case) as the exact solution. But this expression is again obtained by using the flawed choice of w y = w z = 0 (see four lines below Equation (86), p. 5339). This mistaken procedure leads to a critical exponent beta = 3 / 8 for the magnetization of the 3D model. But it also gives the same exponent 3/8 for the 2D model – since Equation (99) reduces to 2D by setting K ’’ = 0 or x 4 = 1 : This is clearly wrong, since we know from the exact solution of Yang that the 2D exponent is 1/8. Moreover, the expansion of the expression (102), namely, 1 – 6 x^8 - 12 x^{10} - 18 x^{12} - .. (see Equation (103)), fails to agree with the exact low-temperature expansion of the spontaneous magnetization of the simple cubic lattice , which is 1 – 2 x^6 - 12 x^{10} + 14 x^{12} - … : A cardinal, golden rule for verifying the validity of any proposed exact solution is that it must yield, term by term, the correct high and low temperature expansions. Indeed, in many cases, including, in particular, the case of the three-dimensional Ising ferromagnet, this is the subject of a mathematical theorem (see Sinai ). Since the author clearly realizes that his conjectured expressions fail in this test, he has assembled a variety of reasons to justify the failure. He states that the test works in d = 2 dimensions because ‘‘ in the 2D case, we are extremely lucky because both the high- and low-temperature expansions are exact and convergent ’’ (section 8.2.3, p.5382, 13th line in second paragraph). To explain the failure of the conjectured free energy, for example, the author argues that the known exact high-temperature expansion holds only ‘‘ at/near ’’ infinite temperature (see line 1, p.5331 and four lines below Equation (A13), p.5406) and thus for finite temperatures one must use the weights w y = w z = 0. This argument of arbitrarily dividing ‘‘ at/near infinite ’’ and ‘‘ finite temperatures ’’ is flawed. Indeed, the suggestion contradicts general rigorous results establishing the finite radii of convergence of the high- T and low- T expansions and their exact representation of the thermodynamic limit for all d = 2 . To explain the incorrect prediction of beta = 3 / 8 for the 2D spontaneous magnetization, the author argues in section 4.2 that there exists a certain region in the interaction parameter space where the exponent beta crosses over from the 3D value 3/8 to the 2D value 1/8. This suggestion is contrary to well-established understanding of critical phenomena and crossover behavior and is, thus, implausible. To patch up the disagreement of (102) with the exact low-temperature expansion, the author states as his opinion ‘‘ that the requirement, Philosophical Magazine 3095 where the exact expression must be equal, term by term, to the so-called exact low-temperature expansion has, for a long time, reflected a pious hope ’’ (see the first paragraph of section 8.2.2, p.5377). This opinion, as noted above, contradicts a host of long established rigorous results for Ising and more general models . In summary, Zhang’s suggestion that the free energy be expressed as a four-fold integral has not produced a solution to the 3D Ising model. Specifically, the conjectured expressions (74) and (102), in which the crucial temperature-dependent weights w y and w z have been set to zero, cannot be exact solutions. Furthermore, the arguments advanced for this step are unsupported and, hence, carry no conviction. In conclusion, the various conjectured relations for the value of T c , for critical exponents, etc., including others not discussed in this note (such as the true range of correlation in section 5.4) are false. Acknowledgement We are grateful to Professor Michael Aizenman for advice in connection with the reference listed below under . References E. Ising, Z. Phys. 31 (1925) p.253. L. Onsager, Phys. Rev. 65 (1944) p.117. C.N. Yang, Phys. Rev. 85 (1952) p.808. Z.-D. Zhang, Phil. Mag. 87 (2007) p.5309. A.J. Guttmann and I.G. Enting, J. Phys. A 26 (1993) p.807. See, for example, J.W. Essam and M.E. Fisher, J. Chem. Phys. 38 (1963) p.802, Appendix A. Ya. G. Sinai, Theory of Phase Transitions : Rigorous Results , Pergamon Press, Oxford, 1982, Chapter II; J. Glimm and A. Jaffe, Quantum Physics , 2nd ed. Springer, New York, 1987, Chapters 18, 20, et seq .; R.B. Israel, Commun. Math. Phys. 50 (1976) p.245; M. Zahradnik, J. Stat. Phys. 47 (1987) p.725. Philosophical Magazine Vol. 88, No. 26, 11 September 2008, 3103 Rejoinder to the Response to ‘Comment on a recent conjectured solution of the three-dimensional Ising model’ F.Y. Wu a , B.M. McCoy b , M.E. Fisher c * and L. Chayes d a Department of Physics, Northeastern University, Boston, USA; b C.N. Yang Institute for Theoretical Physics, State University of New York, New York, USA; c Institute for Physical Science and Technology, University of Maryland, MD, USA; d Department of Mathematics, University of California, Los Angeles, USA ( Received 30 September 2008; final version received 3 October 2008 ) We add here a few sentences concerning the author’s Response to our Comment criticizing his original claims regarding his conjectured solution of the three-dimensional Ising model . First, we stand by our summary in , where the main purpose was to refute claims made in on the basis of a putative 4-dimensional integral representation. In summarizing his rebuttal, Professor Zhang now admits that ‘‘more research’’ is needed. He goes on, however, to assert that ‘‘the correct reproduction of the high-temperature expansion cannot be a coincidence.’’ We consider this remark to be quite misleading: indeed, we point out in that the reproduction of the high- T series in is merely a fit of 11 unknown expansion coefficients (for the weights w y and w z ) to ensure agreement with the 11 exactly known high- T terms. Notably, no further high- T series coefficients are proposed in ; however, since this fit turns out to play no further role, it remains true that the conjectured solution does not reproduce the exact high- T expansion. We do not find the majority of the issues addressed in the Response to be relevant to our disproof of , which also stressed the failure of the conjectured solution to generate the correct low- T expansions. In our view, a refusal to accept the conclusions of the rigorous work (cited in ) for the applicability of the long-known expansions – at high enough and low enough T – to the exact solution for the thermodynamic limit, constitutes a denial of the mathematical basis of statistical mechanics. References Z.-D. Zhang, Phil. Mag. 88 (2008) p.3097. F.Y. Wu, B.M. McCoy, M.E. Fisher et al., Phil. Mag. 88 (2008) p.3093. Z.-D. Zhang, Phil. Mag. 87 (2007) p.5309. _____________________________ *Corresponding author. Email: xpectnil@umd.edu
Dear Professor Wang: I thought it would add value to your blog on the 3D Ising fiasco, if you could post a recent email by Professor Michael Fisher and myself. In January 2008, Dr. Z.D. Zhang emailed 100 or so prominent physicists around the world announcing the publication of his article on the 3D Ising model in the journal Philosophical Magazine (PM), and attached an electronic copy of the reprint with that email. Since some among the recipients may not be aware of the subsequent PM publications of the Comments and Rejoinders by myself, Fisher, and others, Professor Fisher and I have very recently, on March 12, 2009, sent an email to those who received Dr. Zhang's email, to call their attention to the subsequent publications and our view on the value of the Zhang article. I am attaching the email by Professor Fisher and myself below - I was able to get hold of Dr. Zhang's recipient list since I was on the receiving end of his email. Please feel free to post the 3 attachments also, if they have not appeared previously on any of the relevant blogs. With regards, F. Y. Wu _______________________________________ Dear Colleague: You may recall in January last year you received an email from Z.D. Zhang attaching a 111-page paper he published in the Philosophical Magazine (PM), presenting his "conjectured exact solution " of the 3D Ising model. It is sad and unfortunate that PM chose to publish the Zhang paper. In our opinion, the paper is without any merit or value;and, as far as we can tell, the manuscript had previously received serious, strongly negative reviews explaining its deficiencies from a number of journals. After long consideration and at the invitation of PM, wesubmitted a Comment on Zhang's paper. The Comment was written in collaboration with Barry McCoy and Lincoln Chayes.Our Comment was followed by a brief Rejoinder in response to Zhang’s quite lenghthy Reply. We are attaching both our Comment and Rejoinder for your infomation. Also attached is a PM Erratum correcting some editorial errors. Zhang’s Reply to our Comment appeared on pp. 3097-3101 of the same PM issueand can also befound atarXiv 0812.2330, if you are interested. We have since learned that Jacques Perk also commented on Zhang’s paper. His Comment and subsequent Rejoinderare to appearin PM Vol. 89, 761-764,769-770 (2009) (also at arXiv:0811.1802; arXiv:0901.2935 and its updates). We sincerely hope that with these publications the saga of the Zhang conjecture can be put to rest. With regards, Fred Y. Wu and Michael E. Fisher
有 Potsdam 和 新abc 在几处留言， 对本人开展猜想工作的经费资助提出质疑。本人特此声明如下： 本人在开展猜想工作时没有拿到国家一分钱是事实 , 但这不影响本人在猜想原文中致谢使我能活命的基金资助(所有的项目均按时圆满完成,与猜想无关) 。 本人主要是从感恩的心态出发 . 在猜想原文中致谢基金如下 : The author acknowledges the continuous support of the National Natural Science Foundation of China since 1990 (under grant numbers 59001452, 59371015,19474052, 59421001, 59725103, 59871054, 59831010, 50171070, 10274087,10674139, 50331030, and 50332020) and the support of the Sciences and Technology Commission of Shenyang since 1994. 另外， 在最近的答伍法岳等人和Perk教授的答复意见文章中我感谢中国国家自然科学基金委员会的资助（项目号： 50831006, 10674139 和 50331030 ）。猜想是在 2007 年底发表的 , 可以看出仅 50831006 是 2008 年申请成功 , 项目的 题目和研究内容与猜想也无关 . 在这里致谢的目的仍然是感恩 ! 当然 , 由于猜想的严格证明是个长期的工作 , 我希望以后猜想的工作能够得到国家的支持 !
标签: 猜想
