DOCSLIB.ORG

Supersymmetry in Stochastic Processes with Higher-Order Time

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Supersymmetry in Stochastic Processes with Higher-Order Time Supersymmetry in Stochastic Processes with Higher-Order Time Derivatives Hagen KLEINERT and Sergei V. SHABANOV ∗ † Institut f¨ur Theoretische Physik, Freie Universit¨at Berlin, Arnimallee 14, 14195 Berlin, Germany The supersymmetry implies QS = 0. A supersymmetric path integral representation is devel- The determinant (3) should not be confused with the oped for stochastic processes whose Langevin equation con- partition function for fermions governed by the Hamil- tains any number N of time derivatives, thus generalizing the tonian associated with the action (4). Instead of a Langevin equation with inertia studied by Kramers, where trace over external states it contains only the vacuum- N = 2. The supersymmetric action contains N fermion fields to-vacuum transition amplitude for the imaginary-time with first-order time derivatives whose path integral is evalu- ated for fermionless asymptotic states. interval under consideration. In the coherent state rep- resentation,c ¯t and ct are set to zero at the initial and 1. For a stochastic time-dependent variable xt obeying a first-order Langevin equation final times, respectively [2]. The path integral (2) can also be rewritten in a canon- ical Hamiltonian form by introducing an auxiliary Gaus- Lt[x] ≡ x˙ t + F (xt)= ηt, (1) sian integral over momentum variables pt, and replac- H 2 driven by a white noise ηt with hηti = 0, hηtηt′ i = δtt′ , ing Sb by Sb = dt (pt /2 − iptLt). The generator the correlation functions hx ··· x i can be derived from of supersymmetry for the canonical action is QH = t1 tn R a generating functional dt (ictδxt + ptδc¯t ). This form has an important advan- tage to be used later that it does not depend explicitly R i dtJx −Sb+i dtJx on D , so that the above analysis remains valid also for Z[J]= he i = Dx ∆ e , (2) t more general colored noises with an arbitrary correlation R Z R function hηatηbt′ i = (Dab)tt′ 6= δtt′ . 1 2 with an action Sb = 2 dt Lt , and a Jacobian ∆ = Inserting (3) into (2), the generating functional be- det δx ′ Lt. We denote by δx ′ Lt the functional deriva- comes t R t tive of Lt[x] with respect to its argument. Explicitly: H ′ ′ i dtJx −S −Sf +i dtJx δxt′ Lt = [∂t + F (xt)]δtt . The time variable is written as Z[J]=he i= DpDxDc¯Dc e . (6) a subscript to have room for functional arguments after R Z R a symbol. It was pointed out by Parisi and Sourlas [1] that by expressing the Jacobian ∆ as a path integral over This representation makes the stochastic process (1) Grassmann variables equivalent to by supersymmetric quantum mechanical system in imaginary time. In the supersymmetric formu- lation of a stochastic process, there exists an infinity of ∆= Dc¯Dce−Sf (3) Ward identities between the correlation functions which Z can be collected in the functional relation with a fermionic action −SH H arXiv:quant-ph/9705042v1 24 May 1997 DpDxDc¯Dce Q Φ=0, (7) ′ ′ S = dtdt c¯ δ ′ L c ′ = dt c¯ (∂ + F )c , (4) f t xt t t t t t Z Z Z valid for an arbitrary functional Φ ≡ Φ[p, x, c,c¯ ]. The the combined action S ≡ Sb+Sf becomes invariant under Ward identities simplify a perturbative computation of supersymmetry transformations generated by the nilpo- the correlation functions. 2 tent (Q = 0) operator A proof of the equivalence of (1) to (2) requires a regu- larization of the path integral, most simply by time slic- Q = dt (icδx − iLtδc¯) (5) ing. This is not unique, since there are many ways of dis- Z cretizing the Langevin equation (1). If one sets ti = iǫ, for i =0, 1, 2,...,M, xi = xti , and Fi = F (xi), then the velocityx ˙ may be approximated by (xi − xi−1)/ǫ. On the sliced time axis, the force F (xt) may act at any time ∗Humboldt fellow; on leave from Laboratory of Theoretical within the slice (ti,ti−1), which is accounted for by a pa- Physics, JINR, Dubna, Russia. † rameter a and a discretization F → aFi + (1 − a)Fi−1. Email: [email protected]; [email protected] Note that the discretized Langevin equation is assumed berlin.de; URL: http://www.physik.fu-berlin.de/˜kleinert. to be causal, meaning that given the initial value of Phone/Fax: 0049/30/8383034 the stochastic variable x0 and the noise configurations 1 η0, η1, ..., ηM−1, the Langevin equation uniquely deter- higher time derivatives mines the configurations of the stochastic variable at later time, x1, x2, ..., xM . The simplest choice of the right- Lt = γ(∂t)x ˙ t + F (xt)= ηt , (9) hand side of the Langevin equation compatible with the where γ is a polynomial of any order N − 1, thus pro- causality is to set it equal to ηi−1. In general, one can M ducing N time derivatives on xt. This Langevin equa- replace ηi−1 by 1 Aj−1,i−1ηi−1 with A being an or- thogonal matrix, AT A = 1. The latter is just an evi- tion may account for inertia via a term m ∂t in γ(∂t), dence of the symmetryP of the stochastic process with the and/or an arbitrary nonlocal friction dτγτ x˙ t−τ = N−1 γ ∂+1x where γ = dτγ (−τ)n/n!. The main white noise with respect to orthogonal transformations n=0 n t t n τ R η → (Aη) . problem is to find a proper representation of the more t t P R Some specific values of the interpretation parameter complicated determinant ∆ = det δxt′ Lt in terms of a have been favored in the literature, with a = 0 or Grassmann variables. The standard formula (3), though 1/2 corresponding to the so-called Itˆo- or Stratonovich- formally applicable, does not provide a proper represen- related interpretation of the stochastic process (1), re- tation of the determinant of an operator with higher- spectively [2,3]. In the time-sliced path integral, these order derivatives because of the boundary condition values correspond to a prepoint or midpoint sliced ac- problem. This problem is usually resolved via an op- tion [2,4]. Emphasizing the a-dependence of the sliced erator representation of the associated fermionic system. H In the stochastic context it has so far been discussed only action, we shall denote it by Sa . This action is super- H H for the single time derivative [2]. In the first-order for- symmetric for any a: Q Sa = 0, and the sliced genera- H malism, the fermion path integral can be defined in terms tor Q = i(ici∂xi +pi∂c¯i ) turns out to be independent of both the interpretation parameter a and the width ǫ of coherent states [2] with the above discussed vacuum- of time slicingP [2]. A shift of a changes the action by the to-vacuum boundary conditions. Higher-derivative theo- Q-exact term, ries, however, have many unphysical features, in partic- ular states with negative norms [6,7], and it is a priori H H H Sa+δa = Sa + δaQ G, (8) unclear how to define the boundary conditions for the as- sociated fermionic path integral. In gauge theories, the where G is a function of a and a functional of p, x, c,¯ c. Faddeev-Popov ghosts give an example of a fermionic the- This makes the Ward identities independent of a, i.e. ory with higher-(second-)order derivatives. There, un- on the interpretation of the Langevin equation. In- physical consequences of the negative norms of the ghost − H − H H deed, setting δa = −a we find e Sa = e S0 eaQ Φ ≡ states are avoided by imposing the so called BRST invari- H −S0 H ′ e (1 + Q Φa). Substituting this relation into (7) we ant boundary conditions upon the path integral. For the observe that the a-dependence drops out from the Ward above stochastic determinant with higher-order deriva- H H identities because of the supersymmetry Q S0 = 0 and tives, the correct boundary condition are unknown. the nilpotency (QH )2 = 0. 2. The solution proposed by us in this work is best The simplest situation arises for the Itˆochoice, a = 0. illustrated by first treating Kramers’ process where one Then the sliced fermion determinant ∆ becomes a triv- more time derivative is present, accounting for particle ial constant independent of x. In the continuum limit inertia, i.e. where γ(∂ )= ∂ + γ for a unit mass m ≡ 1. of the path integral, however, this choice is inconvenient t t Omitting the time subscript of the stochastic variables, since then the limiting action S cannot be treated as an 0 for brevity, we replace the stochastic differential equation ordinary time integral over the continuum Lagrangian. (9) by two coupled first-order equations Instead, Sa=0 goes over into a so-called Itˆostochastic in- tegral [2]. The Itˆointegral calculus [3] differs in several Lv =v ˙ + γv + F (x)= νv, (10) respects form the ordinary one, most prominently by the property dx 6= dtx˙. This difficulty is avoided taking Lx =x ˙ − v = νx, (11) the Stratonovich value a = 1/2, for which the contin- There are now two independent noise variables, which uum limitR of S Ris an ordinary integral [2,5]. Splitting 1/2 fluctuate according to the path integral (8) as Sa = S1/2 + (a − 1/2)Q G, the non-Stratonovich part vanishes in the continuum limit because Q does not depend on the slicing parameter ǫ, whereas G is pro- hF [x, v]i = DνxDνvF [x, v] portional to ǫ → 0 [2]. For a = 1/2, formula (6) has Z 2 2 −1/2 dt[ν /2(1−σ)+(ν ˙ +γν ) /2σ] a conventional continuous interpretation as a sum over ×e v x x .
Load more

Recommended publications
  • Field Theoretical Approaches to the Superconducting Phase Transition 3
    October 23, 2018 7:17 WSPC/Trim Size: 9in x 6in for Review Volume rv-nogueira CHAPTER 1 Field Theoretical Approaches to the Superconducting Phase Transition Flavio S. Nogueira and Hagen Kleinert Institut f¨ur Theoretische Physik, Freie Universit¨at Berlin, Arnimallee 14, D-14195 Berlin, Germany E-mail: [email protected] Several field theoretical approaches to the superconducting phase tran- sition are discussed. Emphasis is given to theories of scaling and renor- malization group in the context of the Ginzburg-Landau theory and its variants. Also discussed is the duality approach, which allows to access the strong-coupling limit of the Ginzburg-Landau theory. 1. Introduction The Ginzburg-Landau (GL) phenomenological theory of superconductivity1 was proposed long before the famous Bardeen-Cooper-Schrieffer (BCS) microscopic theory of superconductivity.2 A few years after the BCS theory, Gorkov3 derived the GL theory from the BCS theory. This has been done by deriving an effective theory for the Cooper pairs valid in the neighborhood of the crit- ical temperature Tc. The modern derivation proceeds most elegantly via functional integrals, introducing a collective quantum field ∆(x,t) for the Cooper pairs via a Hubbard-Stratonovich transformation and integrating out the fermions in the partition function.4 Amazingly, the GL theory has kept great actuality up to now. It is arXiv:cond-mat/0303485v1 [cond-mat.supr-con] 22 Mar 2003 highly relevant for the description of high-Tc superconductors, even though the original BCS theory is inadequate to treat these materials. The success of the GL theory in the study of modern problems of superconductivity lies on its universal effective character, the details of the microscopic model being unimportant.
    [Show full text]
  • PARTICLES and QUANTUM FIELDS Particles and Quantum Fields
    PARTICLES AND QUANTUM FIELDS Particles and Quantum Fields Hagen Kleinert Professor of Physics Freie Universit¨at Berlin To my wife Annemarie and our son Hagen Michael II Preface This book arose from lectures I gave at the Freie Universit¨at Berlin over the past five decades. They were intended to prepare graduate students for their research in elementary-particle physics or in many-body theory of condensed matter. They should serve as a general introduction and a basis for understanding more advanced work on the subject. The theory of quantum fields presented in this book is mainly based on the perturbative approach. Elementary particles are introduced initially without any interactions. These are added later, and their strength is parametrized by some coupling constant g. The consequences are studied order by order in g, with the particles propagating forward from interaction to interaction. Such a treatment is clearly a gross simplification of what happens in nature, where even the existence of a free particle involves the full interaction from the very beginning. Nevertheless, this kind of procedure has been the basis of many successful theories. In all of them, there exist dominant freely propagating excitations or elementary particles at least in some experimentally accessible limit. The most prominent example is the theory of strongly interacting particles. There they are described as being composed of quarks held together by gluons which interact via a nonabelian gauge theory called quantum chromodynamics (QCD). In the limit of large energies, the particles behave like free point-like particles. This behavior was named parton-like by Richard Feynman.
    [Show full text]
  • Applied Quantum Field Theory
    Applied Quantum Field Theory Contents 1 Topics 927 2 Participants 929 2.1 ICRANet participants . 929 2.2 Ongoing collaborations . 929 2.3 Students . 929 3 Brief Description 933 4 Publications (2005 - 2009) 935 Bibliography 947 925 1 Topics • Pair Creation in Strong Inhomogeneous Electric Fields – Semi-classical description of pair production – WKB transition probability for Klein-Gordon Field – Rates of pair production – Applications ∗ Constant electric field ∗ Sauter electric field ∗ Sauter electric field – Tunneling into Bound States – Coulomb electric field ∗ WKB transition probability ∗ Semiclassical quantization for point-like nucleus ∗ Semiclassical quantization for finite-size nucleus • Theory of Phase Transitions – Superfluid-Normal Transition from condensation of vortex lines – Extending the London theory of Superconductors to full hydrody- namics with vortices • Bose Einstein Condensation – Bose-Einstein Condensates with Long-Range Interactions – Variational Resummation of the Effective Potential in f4-Theory – Thermodynamic Properties of F=1 Spinor Bose-Einstein Conden- sates – Ultracold Dilute Boson-Fermi-Mixtures – Bosonen im Optischen Gitter – Spinor-Bosons in Cubic Optical Lattices – Bose-Einstein-Kondensation in Kanonischen Ensembles – Thermodynamische Eigenschaften von Bose-Gasen 927 1 Topics • Gravity from Spacetime Defects in Universe – Deriving Einstein-Cartan geometry from multivalued coordinate transformations and local rotations – Reformulating Solutions of Einstein equations in teleparallel space- time • Econophysics – Measuring Market Temperatures – Hedging against Non-Gaussian Fluctuations 928 2 Participants 2.1 ICRANet participants • Hagen Kleinert • Remo Ruffini • She-Sheng Xue 2.2 Ongoing collaborations • Alexander Chervyakov (FU-Berlin, Germany) • Patrick Haener (Nomura Bank, London, Great Britain) • Petr Jizba (FU-Berlin, Germany) • Flavio Nogueira (FU-Berlin, Germany) • Axel Pelster (Universitat¨ Essen-Duisburg, Germany) 2.3 Students • Tim X.J.
    [Show full text]
  • Tricritical Point in Quantum Phase Transitions of the Coleman-Weinberg Model at Higgs Mass
    Tricritical Point in Quantum Phase Transitions of the Coleman-Weinberg Model at Higgs Mass Miguel C N Fiolhais∗ LIP, Department of Physics, University of Coimbra, 3004-516 Coimbra, Portugal Hagen Kleinerty Freie Universit¨atBerlin, Institut f¨urTheoretische Physik Arnimallee 14, D-14195 Berlin, Germany (Dated: June 28, 2013) Abstract The tricritical point, which separates first and second order phase transitions in three-dimensional superconductors, is studied in the four-dimensional Coleman-Weinberg model, and the similarities as well as the differences with respect to the three-dimensional result are exhibited. The position of the tricritical point in the Coleman-Weinberg model is derived and found to be in agreement with the Thomas-Fermi approximation in the three-dimensional Ginzburg-Landau theory. From this we deduce a special role of the tricritical point for the Standard Model Higgs sector in the scope of the latest experimental results, which suggests the unexpected relevance of tricritical behavior in the electroweak interactions. The following paper is published in Phys. Lett. A: http://www.sciencedirect.com/science/article/pii/S0375960113005768 INTRODUCTION this way, the GL theory has become what may be called the Standard Model of superconductive phenomenology. Ever since the formulation in 1964 of the electroweak A similar Ginzburg-Landau-like scalar field theory spontaneous symmetry breaking mechanism [1{5] to ex- with quartic interaction is successful in unifying the weak plain how elementary particles acquire mass, supercon- and the electromagnetic interactions, so that it has be- ductivity and high-energy physics became intimately con- come the Standard Model of particle physics, also called nected.
    [Show full text]
  • Order of Superconductive Phase Transition
    1 Condensed Matter Physics, 2005, Vol. 8, No. 1(41), pp. 1–12 Order of superconductive phase transition H.Kleinert 2 Institut f¨ur Theoretische Physik, Freie Universitat¨ Berlin, Arnimallee 14, D–14195 Berlin, Germany Received December 26, 2004 3 On the occasion of Reinhard Folk’s 60th birthday, I give a brief review of 4 the theoretical progress in understanding the critical properties of super- 5 conductors. I point out the theoretical difficulties in finding a second-order 6 transition in the Ginzburg-Landau Model with O(N)-symmetry in 4 − ε Di- 7 mensions, and the success in predicting the existence and location of a 8 tricritical point with the help of a dual disorder theory. 9 Key words: 10 PACS: 11 1. Introduction 12 Among the many important contributions made by Reinhard Folk in the field 13 of critical phenomena is his work on the properties of the superconducting phase 14 transitions [1,2], where he discussed the effect of two-loop contributions on the renor- 15 malization flow. To point out its significance, let me briefly recall the historic back- 16 ground of the problem studied by him with various collaborators, most prominently 17 with Yurij Holovatch. 18 Until thirty years ago, the superconductive phase transition was always assumed 19 to be of second order. All critical exponents had the mean-field values required by 20 the Ginzburg-Landau model. In 1974, however, it was noted by Halperin, Lubensky, 21 and Ma [3] (see also [4]) that the renormalization group treatment of the Ginzburg- 22 Landau model of superconductivity did not produce a critical point.
    [Show full text]
  • Mathematical Aspects of the Feynman Path Integral
    Bachelor's thesis for mathematics and physics Rijksuniversiteit Groningen Faculty of Science and Engineering Mathematical aspects of the Feynman path integral Mathematics supervisor: Dr. M. Seri Author: Daniel W. Boutros Physics supervisor: Prof. Dr. D. Boer July 12, 2020 Abstract Path integration methods are of crucial importance to quantum mechanics and quantum field theory. There are multiple ways the path integral can be constructed, one method uses the link between the Fokker-Planck and the Langevin equations, as is covered in this thesis. This is related to the standard derivation of the path integral in physics, in which the time is discretised. We study the mathematical problems related to the Feynman path integral, in particular the impossibility of a Lebesgue-type measure on the space of paths. It is discussed how oscillatory integrals can be used to have a well-defined `integral' on an infinite dimensional space. This formalism is subsequently applied to both the harmonic and the anharmonic oscillator. We prove an infinite dimensional oscillatory integral exists and obtain a convergent expression for both these cases, under conditions. Examples of these conditions are the initial wavefunctions being Schwartz functions as well as conditions on the endtime, angular frequency and coupling constant. Finally, a possible approach to establish an oscillatory integral for the hydrogen atom is discussed. It is proven that the result is no longer independent of the sequence of projection operators, which is a key step towards a rigorous path integral for this system. Contents 1 Mathematical introduction4 2 Physical introduction9 3 From the Fokker-Planck equation to the path integral 12 3.1 Revision of probability theory.........................
    [Show full text]
  • Kleinert Hagen
    Kleinert Hagen Position: Professor of Theoretical Physics Sub-Project Applied Quantum Field Theory Period covered:1964-2007 I Scientific Work 1. Theory of Defect-Induced Phase Transitions 2. Quark and String Physics 3. Quantum Mechanics 4. Classical and Statistical Mechanics in Spaces with Curvature and Torsion 5. Classical Statistics 6. Quantum Statistics 7. Polymer Physics 8. Field Theory of Liquid Crystals 9. Fluctuation Effects in Membranes 10. Superfluid Helium 3 11. Superconductivity 12. Mathematical Physics 13. Stochastic Physics 14. Supersymmetry in Nuclear Physics 15. Financial Markets II Conferences and educational activities Conferences and Other External Scientific Work Hundreds of conferences. Work With Students Hundreds of students. Diploma thesis supervision Hundreds of theses. Other Teaching Duties All courses of Theoretical Physics Work With Postdocs Many Postdocs financed by FU-Berlin, Humboldt Foundation, DFG, and DAAD. III Service activities Within ICRANet Lecturing and exchange of ideas. Organization of big Marcel-Grossmann Conference 2006. Research and collaborations. Outside ICRANet Teaching and research, writing textbooks. IV Other Study and Degrees: 1960 -1963: TH Hannover; there 1962 -BS with fist class honors 1963 -1964: Georgia Institute of Technology, Atlanta, Georgia, USA; there 1964 -Master of Science Fall 1964: Washington University, St. Louis, USA Spring 1965: University of Wisconsin, Madison, USA 1965 -1967: University of Colorado, Boulder, Colorado, USA; there 1967 -Ph. D. Spring 1969: Habilitation at the Free University Berlin Positions: Fall 1963 – Research Assistant at EURATOM, Ispra, Italy June 1967 – Research Associate at the University of Colorado Jan. 1968 – Assistant Professor at the University of Montana Oct. 1969 – Associate Professor at the Free University Berlin Okt.
    [Show full text]
  • String Theory”
    EJTP 8, No. 26 (2011) 27–34 Electronic Journal of Theoretical Physics The Purely Geometric Part of “Dark Matter”– A Fresh Playground for “String Theory” Hagen Kleinert∗ Institut f¨ur Theoretische Physik, Freie Universit¨at Berlin, 14195 Berlin, Germany ICRANeT Piazzale della Repubblica, 10 -65122, Pescara, Italy Received 24 September 2011, Accepted 4 November 2011, Published 17 January 2012 Abstract: We argue that part of “dark matter” is not made of matter, but of the singular world-surfaces in the solutions of Einstein’s vacuum field equation Gμν = 0. Their Einstein- Hilbert action governs also their quantum fluctuations. It coincides with the action of closed bosonic “strings” in four spacetime dimensions, which appear here in a new physical context. Thus, part of dark matter is of a purely geometric nature, and its quantum physics is governed by the same string theory, whose massless spin-2 particles interact like the quanta of Einstein’s theory. c Electronic Journal of Theoretical Physics. All rights reserved. Keywords: Dark Matter; General Relativity; Einstein Field Equations; String Theory; Cosmology PACS (2010): 95.35+d; 04.60.-m; 04.20.-q;04.60.Cf; 98.80.-k Dark matter was postulated by F. Zwicky in 1933 to explain the “missing mass” in the orbital velocities of galaxies in clusters. Later indications came from measurements of the orbital motion of stars inside a galaxy, where a plot of orbital velocities versus distance from the center was attributed to large amounts of invisible matter. The Friedmann model of the evolution of the universe indicates that dark matter constitutes a major percentage of the mass energy of the universe, and there are many speculations as to its composition.
    [Show full text]
  • Kleinert Hagen
    Kleinert Hagen Position: Richard Feynman Professor Period covered: 2009 2010 List of Publications H. Kleinert, From Landau's Order Parameter to Modern Disorder Field Theory in L.D. Landau and his Impact on Contemporary Theoretical Physics, Horizons in World Physics 264 (2008), A. Sakaji and I. Licata (preprint). P. Jizba, H. Kleinert, and P. Haener Perturbation Expansion for Option Pricing with Stochastic Volatility Physica A 388 (2009) 3503 (preprint) P. Jizba, H. Kleinert, and F. Scardigli Uncertainty Relation on World Crystal and its Applications to Micro Black Holes (arXiv:0912.2253) Phys. Rev. D 81, 084030 (2010) H. Kleinert New Gauge Symmetry in Gravity and the Evanescent Role of Torsion (arxiv/1005.1460) EJTP 24, 287 (2010) H. Kleinert Converting Divergent Weak-Coupling into Exponentially Fast Convergent Strong-Coupling Expansions (arXiv:1006.2910) preprint (2010) Petr Jizba and Hagen Kleinert Superstatistics approach to path integral for a relativistic particle (arxiv/1007.1007.3922) Phys. Rev. D 82, 085016 (2010) (2010) II Conferences and educational activities Work With Students • Tim X.J. Chen (FU-Berlin, Germany) • Konstantin Glaum (FU-Berlin, Germany) • Sonja Overesch (FU-Berlin, Germany) • Walja Korolevski (FU-Berlin, Germany) • Mathias Ohlinger (FU-Berlin, Germany) • Moritz Sch ¨ utte (FU-Berlin, Germany) • Steffen R¨othel (FU-Berlin, Germany) • Matthias Ohliger (FU-Berlin, Germany) • Pascal Mattern (FU-Berlin, Germany)1469 • Ednilson Santos (FU-Berlin, Germany) • Alexander Hoffmann (FU-Berlin, Germany) • Parvis Soltan-Panahi:
    [Show full text]
  • Frontiers of Quantum and Mesoscopic Thermodynamics 28 July - 2 August 2008, Prague, Czech Republic
    Frontiers of Quantum and Mesoscopic Thermodynamics 28 July - 2 August 2008, Prague, Czech Republic Satellite of the 22nd General Conference of the EPS Condensed Matter Division 25 - 29 August 2008, Roma Under the auspicies of Prof. RNDr. Vaclav´ Paces,ˇ DrSc. President of the Academy of Sciences of the Czech Republic Prof. RNDr. Vaclav´ Hampl, DrSc. Rector of the Charles University Supported by • Condensed Matter Division of the European Physical Society • Committee on Education, Science, Culture, Human Rights and Petitions of the Senate of the Parliament of the Czech Republic • Institute of Physics, v. v. i., Academy of Sciences of the Czech Republic • Faculty of Mathematics and Physics, Charles University • Center for Theoretical Study, Charles University and the Academy of Sciences of the Czech Republic • Institute for Theoretical Physics, University of Amsterdam Topics • Quantum, mesoscopic and classical thermodynamics • Foundations of quantum physics • Physics of quantum measurement • Quantum optics • Quantum dissipation, decoherence and noise • Physics of quantum computing and quantum information • Non-equilibrium quantum statistical physics • Macroscopic quantum behavior • Mesoscopic and nanomechanical systems • Spin systems and their dynamics • Brownian motion, molecular motors, ratchet systems, rectified motion • Physics of biological systems • Relevant experiments from the nano- to the macro-scale Scientific committee Chair: Theo Nieuwenhuizen (Amsterdam) Co-Chair: Vaclav´ Spiˇ ckaˇ (Prague) Roger Balian (CEA-Saclay) Gordon Baym
    [Show full text]
  • Collective Quantum Fields in Plasmas, Superconductors, and Superfluid 3He, Collective Quantum Fields in Plasmas, Superconductors, and Superfluid 3He
    Collective Quantum Fields in Plasmas, Superconductors, and Superfluid 3He, Collective Quantum Fields in Plasmas, Superconductors, and Superfluid 3He Hagen Kleinert Professor of Physics Freie Universit¨at Berlin Preface Under certain circumstances, many-body systems behave approximatly like a gas of weakly interacting collective excitations. Once this happens it is desirable to replace the original action involving the fundamental fields (electrons, nucleons, 3He, 4He atoms, quarks etc.) by another one in which all these excitations appear as explicit independent quantum fields. It will turn out that such replacements can be performed in many different ways without changing the physical content of the theory. Sometimes, there esists a choice of fields associated with dominant collective excitations displaying weak residual interactions which can be treated perturbatively. Then the collective field language greatly simplifies the description of the physical system. It is the purpose of this book to discuss a simple technique via Feynman path integral formulas in which the transformation to collective fields amounts to mere changes of integration variables in functional integrals. After the transformation, the path formulation will again be discarded. The resulting field theory is quantized in the standard fashion and the fundamental quanta directly describe the collective excitations. For systems showing plasma type of excitations, a real field depending on one space and time variable is most convenient to describe all physics. For the opposite situation in which dominant bound states are formed, a complex field depending on two space and one or two time coordinates will render th emore ecnonomic description. Such fields will be called bilocal. If the potential becomes extremely short range, the bilocal field degenerates into a local field.
    [Show full text]
  • Path Integral Formulation of Quantum Tunneling: Numerical Approximation and Application to Coupled Domain Wall Pinning
    Path Integral Formulation of Quantum Tunneling: Numerical Approximation and Application to Coupled Domain Wall Pinning Matthew Ambrose Supervisor: Prof Robert Stamps Honours Thesis submitted as part of the B.Sc. (Honours) degree in the School of Physics, University of Western Australia Date of submission: 24/October/08 Acknowledgments Many thanks to my supervisor Bob for all his support, teaching me all that he has and generally helping give me direction in my research. Also thanks to those in the condensed matter group who have given a great deal of assistance, in particular Karen Livesy who gave a great deal of her time to proof read the thesis A thanks is also extended to my family and friends who have been incredibly supportive. Abstract In this thesis we investigate numerically the macroscopic tunneling rates of a coupled system of fronts. We show that it is possible to make a new numerical approximation to the symmetric double well potential that agrees with the analytic expression for the ground state energy level splitting. Using this numerical method we then make the appropriate changes to the ground state energy calculation to treat a non symmetric potential. The introduction of an imaginary part to the ground state energy causes a decay away from the metastable side of the potential. We then treat a system that can decay from a metastable state into more than one lower energy configuration. As an example of such a system we consider a multiferroic system in which the magnetic and electric polarisations are coupled. We then proceeded to compute the tunneling rate numerically and compare the behavior of the tunneling rates for the two possible final states.
    [Show full text]