ترغب بنشر مسار تعليمي؟ اضغط هنا

On a semi-classical limit of loop space quantum mechanics

175   0   0.0 ( 0 )
 نشر من قبل Partha Mukhopadhyay
 تاريخ النشر 2012
  مجال البحث
والبحث باللغة English




اسأل ChatGPT حول البحث

Following earlier work, we view two dimensional non-linear sigma model with target space $cM$ as a single particle relativistic quantum mechanics in the corresponding free loop space $cLM$. In a natural semi-classical limit ($hbar=alpha to 0$) of this model the wavefunction localizes on the submanifold of vanishing loops which is isomorphic to $cM$. One would expect that the relevant semi-classical expansion should be related to the tubular expansion of the theory around the submanifold and an effective dynamics on the submanifold is obtainable using Born-Oppenheimer approximation. In this work we develop a framework to carry out such an analysis at the leading order in $alpha$-expansion. In particular, we show that the linearized tachyon effective equation is correctly reproduced up to divergent terms all proportional to the Ricci scalar of $cM$. The steps leading to this result are as follows: first we define a finite dimensional analogue of the loop space quantum mechanics (LSQM) where we discuss its tubular expansion and how that is related to a semi-classical expansion of the Hamiltonian. Then we study an explicit construction of the relevant tubular neighborhood in $cLM$ using exponential maps. Such a tubular geometry is obtained from a Riemannian structure on the tangent bundle of $cM$ which views the zero-section as a submanifold admitting a tubular neighborhood. Using this result and exploiting an analogy with the toy model we arrive at the final result for LSQM.



قيم البحث

اقرأ أيضاً

102 - S. S. Gubser , I. R. Klebanov , 2002
A world-sheet sigma model approach is applied to string theories dual to four-dimensional gauge theories, and semi-classical soliton solutions representing highly excited string states are identified which correspond to gauge theory operators with re latively small anomalous dimensions. The simplest class of such states are strings on the leading Regge trajectory, with large spin in AdS_5. These correspond to operators with many covariant derivatives, whose anomalous dimension grows logarithmically with the space-time spin. In the gauge theory, the logarithmic scaling violations are similar to those found in perturbation theory. Other examples of highly excited string states are also considered.
201 - Jose L. Cortes , J. Gamboa 2020
An approach to study a generalization of the classical-quantum transition for general systems is proposed. In order to develop the idea, a deformation of the ladder operators algebra is proposed that contains a realization of the quantum group $SU(2) _q$ as a particular case. In this deformation Plancks constant becomes an operator whose eigenvalues approach $hbar $ for small values of $n$ (the eigenvalue of the number operator), and zero for large values of $n$ (the system is classicalized).
A recently introduced numerical approach to quantum systems is analyzed. The basis of a Fock space is restricted and represented in an algebraic program. Convergence with increasing size of basis is proved and the difference between discrete and cont inuous spectrum is stressed. In particular a new scaling low for nonlocalized states is obtained. Exact solutions for several cases as well as general properties of the method are given.
It is expected in physics that the homogeneous quantum Boltzmann equation with Fermi-Dirac or Bose-Einstein statistics and with Maxwell-Boltzmann operator (neglecting effect of the statistics) for the weak coupled gases will converge to the homogeneo us Fokker-Planck-Landau equation as the Planck constant $hbar$ tends to zero. In this paper and the upcoming work cite{HLP2}, we will provide a mathematical justification on this semi-classical limit. Key ingredients into the proofs are the new framework to catch the {it weak projection gradient}, which is motivated by Villani cite{V1} to identify the $H$-solution for Fokker-Planck-Landau equation, and the symmetric structure inside the cubic terms of the collision operators.
84 - Takeshi Morita 2021
Out-of-time-order correlator (OTOC) $langle [x(t),p]^2 rangle $ in an inverted harmonic oscillator (IHO) in one-dimensional quantum mechanics exhibits remarkable properties. The quantum Lyapunov exponent computed through the OTOC precisely agrees wit h the classical one. Besides, it does not show any quantum fluctuations for arbitrary states. Hence, the OTOC may be regarded as ideal indicators of the butterfly effect in the IHO. Since IHOs are ubiquitous in physics, these properties of the OTOCs might be seen in various situations too. In order to clarify this point, as a first step, we investigate the OTOCs in one dimensional quantum mechanics with polynomial potentials, which exhibit butterfly effects around the peak of the potential in classical mechanics. We find two situations in which the OTOCs show exponential growths reproducing the classical Lyapunov exponent of the peak. The first one, which is obvious, is using suitably localized states near the peak and the second one is taking a double scaling limit akin to the non-critical string theories.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا