Everything on this webpage may be viewed on a nicer typeset pdf here. We begin by recalling some classical rstvariation formulas bes87. We note that the space of quantum states is naturally equipped with a riemannian metric, which differs from l 2 and l 1 geometries explored previously. Equivariant connected sums of compact selfdual manifolds 719 group t 2 contains many geometrically different subgroups, there are many dif ferent equivariant deformations to study. Pdf from the jordan product to riemannian geometries on.
Fubinistudy metric known as wigners theorem, and, moreover, the evolution on d is isometric with respect to the information metric. A fubinistudy metric is determined up to homothety. In the case of pure and faithful states on bh, this alternative geometrical description clarifies the analogy between the fubinistudy and the. It is a fact that under stereographic projection the round metric corresponds to the fubini study metric.
Pfa an integrals and invariants of singular varieties. Reversing this procedure we then obtain complex structure degenerations. Guided by considerations from the theory of moduli of riemann surfaces, we discover among other results sufficient topological and differentialgeometric conditions for a minimal riemannian immersion of a 2manifold in complex projective space with the fubinistudy metric to be holomorphic. April 8, 20 geometric quantum mechanics is a novel and prospecting approach motivated by the belief that our world is ultimately geometrical. Lecture notes assignments download course materials. Chapter 16 isometries, local isometries, riemannian. Fibres of the abeljacobi map are complex projective spaces, and the first theorem we prove states that near the bradlow limit the moduli space metric restricted to these fibres is a multiple of the fubini study metric. All analytic subvarieties of a complex projective space are in fact algebraic subvarieties and they inherit the kahler manifold structure from the projective space. Toward a definition of complexity for quantum field theory. The fubinistudy metric is, up to proportionality, the unique riemannian metric on that is invariant under the unitary group, which preserves the scalar product. Department of mathematics, university of wisconsin colleges email.
Consider a complex projective space with its fubini study metric. The inner product determines a fubinistudy metric on pv and the idea is to deform continuously in a rather speci c manner the restriction of this metric to pv as to obtain a complex hyperbolic metric, i. In fact, in the absence of noise, the space of quantum states is a complex projective space, which possesses a unique unitarilyinvariant metric tensor called the fubini study metric tensor. On wigners theorem daniel s freed wigners theorem asserts that any symmetry of a quantum system is unitary or antiunitary. Our main tool is the abeljacobi map, which maps \symd\sigma into the jacobian of \sigma. Fubinis theorem is one of the most powerful tools in real analysis. As a consequence it allows the order of integration to be changed in iterated integrals. Furthermore, the use of the braket notation is totally gratuitous and confusing. The article should take a mainstream approach to the topic, and then possibly indicate the importance of the fubini study metric in physics and other areas. Holomorphic embedding of complex curves in spaces of. The article should take a mainstream approach to the topic, and then possibly indicate the importance of the fubinistudy metric in physics and other areas.
Pdf generalized fubinistudy metric and fisher information. The particular normalization on the metric depends on the application. Moreover, an alternative derivation of these riemannian metric tensors in terms of the gns construction associated to a state is presented. It is a fact that under stereographic projection the round metric corresponds to the fubinistudy metric.
We focus on gaussian states which include ground states of free quantum field theories and their approximations encountered in the context of the continuous version of the multiscale entanglement renormalization ansatz. The fubinistudy metric on cpn is a k ahler metric finally, we sketched the notion of holomorphic line bundle, the correspondence with divisor and chern classes. Request pdf quantum geometric tensor fubini study metric in simple quantum system. What is the volume of complex projective space with fubini. The greens function metric of the fubinistudy metric g fs is also known as the burns metric, and is completely explicit, with mass given by. Now given a sequence of hyperk ahler metrics g j2mwhich are volume collapsing, we can rescale the volume. From geometric quantum mechanics to quantum information p. Equipped with a fubinistudy metric, cp n is a symmetric space. This very classical material can be read in di erent books. A pedagogical introduction geometric quantum mechanics is a novel and prospecting approach motivated by the.
The above manipulations deriving the fisher metric from the euclidean metric can be extended to complex projective hilbert spaces. Fubini study metric and einstein constant mathoverflow. Another possible way of doing it is using that this is a kahler manifold. A pedagogical introduction ran cheng department of physics, university of texas, austin, tx 78712, usa. We study certain one parameter deformations of this metric on the complement of an arrangement a finite union of hyperplanes whose levicivita connection is of dunkl typeinteresting examples are obtained from the arrangements defined by finite complex reflection groups. For instance, there is a semi free s 1 in t 2 so one may either consider a semi free sequivariant deformation or a. Every metric gon the complex projective space satis es the inequality stsys 2 cp n. We investigate notions of complexity of states in continuous manybody quantum systems. The notes for lectures 16, 17, and 18 are from the supplementary notes on elliptic operators. Let n be a complex submanifold of a kahler manifold m. The fubinistudy metricdistance has many nice mathematical properties e. The focus on quantum mechanics is totally inappropriate for an article on a standard construction in differential and algebraic geometry. Then 2one should note here that it is not necessary for each cross section of a null set in the product measure to be measurable.
The set of rays, say rh, is the complex projective space. It is interesting to note that good models from the algebraic point of view e. In this short note we give two proofs based on the geometry of the fubinistudy metric. Thus, we have a complete reciprocity between the schrodinger dynamics on. Collapsing of calabiyau metrics and degeneration of complex. This note is an activityoriented companion to the study of real analysis. Additional significance is given to the fibres of the abeljacobi map by our. Request pdf quantum geometric tensor fubinistudy metric in simple quantum system.
A proof of the theorem using the techniques i described above is given in the book calculus on manifolds by michael spivak, which you can probably find somewhere online. This can be proven by some complicated and unenlightening algebraic manipulation. Fubini s theorem 1 fubini s theorem in mathematical analysis fubini s theorem, named after guido fubini, is a result which gives conditions under which it is possible to compute a double integral using iterated integrals. We generalize the fubinistudy metric for mixed states further and arrive at a set of fubinistudy metriccalled. Our proposal for quantifying state complexity is based on the fubini study metric. Cpnis a complex submanifold the restriction of g fsto xis kahler. Collapsing of calabiyau metrics and degeneration of. Zheng, complex di erential geometry, studies in adv.
The proof of fubinis theorem then follows really quite easily, it just relies on definitions of the riemann integral and some manipulation with summations. Chapter 16 isometries, local isometries, riemannian coverings. In mathematics, the fubini study metric is a kahler metric on projective hilbert space, that is, on a complex projective space cp n endowed with a hermitian form. Quantum geometric tensor fubini study metric in simple quantum system. In fact, gromov proved a more general optimal inequality. Related with from geometric quantum mechanics to quantum infor. Consider a complex projective space with its fubinistudy metric.
There is a unique up to a scalar hermitian metric on complex projective space which may be normalized, the fubinistudy metric. In mathematics, the fubinistudy metric is a kahler metric on projective hilbert space, that is, complex projective space cp n endowed with a hermitian form. Our proposal for quantifying state complexity is based on the fubinistudy metric. Product measures fubini and tonelli prakash balachandran department of mathematics duke university november 11, 2008 this lecture is based on g. Starting with a symplectic manifold, a kahler structure appears as a complex quantisationin that we can associate to it a complex vector space hilbert space h x of holomorphic functions.
In the case of pure and faithful states on bh, this alternative geometrical description clarifies the analogy between the fubini study and the bureshelstrom metric tensor. Banach co metric on the smooth locus of the complex analytic variety xc. In this note, we consider plane algebraic curves equipped with the fubinistudy metric from. This should perhaps be no surprise, as the fubinistudy metric provides the means of measuring information in quantum mechanics. It is intended as a pedagogical companion for the beginner, an introduction to some of the main ideas in real analysis, a compendium of problems, are useful in learning the subject, and an annotated reading or reference list. Fubinistudy metric in the geometric interpretation of berrys phase refs.
The notes for lectures 16, 17, and 18 are from the supplementary notes on. Someone asked this question recently and then deleted it, but i still would like to figure out the answer. Variational formulas in this section, we calculate the third variation of perelmans shrinker entropy at the fubinistudy metric. It is intended as a pedagogical companion for the beginner, an introduction to some of the main ideas in real analysis, a compendium of problems, are useful in learning the.
Lecture iv je viaclovsky park city mathematics institute july 19, 20. Quantum geometric tensor fubinistudy metric in simple. Fubinis theorem 1 fubinis theorem in mathematical analysis fubinis theorem, named after guido fubini, is a result which gives conditions under which it is possible to compute a double integral using iterated integrals. From geometric quantum mechanics to quantum infor 1,411 view.
Geometric quantum mechanics is a novel and prospecting approach motivated by the belief that our world is ultimately geometrical. C is a sum of homogeneous terms of even degree, and the degree 2rpart, denoted cr, is equal to p 12. The space, endowed with the fubinistudy metric, is a compact hermitian symmetric space of rank 1. Since the restriction of the riemannian metric of m to n is hermitian and its kahler form is the restriction of the kahler form of m to n, n together with the induced riemannian metric is a kahler manifold as well. On the measurement problem for a twolevel quantum system. For instance, there is a semifree s 1 in t 2 so one may either consider a semifree sequivariant deformation or a. On the other hand, it seems like maybe there should be a purely or almost purely geometric proof of this fact for the following reason. Equivariant connected sums of compact selfdual manifolds.
1166 1256 557 1418 169 218 1489 732 1053 1571 230 424 25 262 687 1170 719 653 1021 1219 57 633 465 526 528 443 1435 112 1074 274 203 1646 451 387 821 300 1375 376 961 817 775 468 617 335 715