Narrow Results

For a general class of boson–fermion Hamiltonians H acting in the tensor product Hilbert space L²(ℝⁿ) ⊗ ∧(ℂ^r) of L²(ℝⁿ) and the fermion Fock space ∧(ℂ^r) over ℂ^r(n, r ∈ ℕ), we establish, in terms of an n-dimensional conditional oscillator measure, a functional integral representation for the trace Tr(F ⊗ z^N_fe^-tH)(F ∈ L^∞(ℝⁿ), z ∈ ℂ∖{0}, t > 0), where N_f is the fermion number operator on ∧(ℂ^r). We prove a Golden–Thompson type inequality for |Tr(F ⊗ z^N_fe^-tH)|. Also we discuss applications to a model in supersymmetric quantum mechanics and present an improved version of the Golden–Thompson inequality in supersymmetric quantum mechanics given by Klimek and Lesniewski ([Lett. Math. Phys.21 (1991) 237–244]). An upper bound for the number of the supersymmetric states is given as well as a sufficient condition for the spontaneous supersymmetry breaking. Moreover, we derive a functional integral representation for the analytical index of a Dirac type operator on ℝⁿ (Witten index) associated with the supersymmetric quantum mechanical model.

This paper is an exposition of the relationship between Witten’s Chern–Simons functional integral and the theory of Vassiliev invariants of knots and links in three-dimensional space. We conceptualize the functional integral in terms of equivalence classes of functionals of gauge fields and we do not use measure theory. This approach makes it possible to discuss the mathematics intrinsic to the functional integral rigorously and without functional integration. Applications to loop quantum gravity are discussed.

The Faddeev–Popov rules for a local and Poincaré-covariant Lagrangian quantization of a gauge theory with gauge group are generalized to the case of an invariance of the respective quantum actions, $S_{(N)}$, with respect to $N$-parametric Abelian SUSY transformations with odd-valued parameters ${\lambda }_p$, $p=1,\dots ,N$ and generators $s_p$: $s_p{s}_q+s_q{s}_p=0$, for $N=3,4$, implying the substitution of an $N$-plet of ghost fields, $C^p$, instead of the parameter, $\xi$, of infinitesimal gauge transformations: $\xi =C^p{\lambda }_p$. The total configuration spaces of fields for a quantum theory of the same classical model coincide in the $N=3$ and $N=4$ symmetric cases. The superspace of $N=3$ SUSY irreducible representation includes, in addition to Yang–Mills fields ${{𝒜}}^{\mu }$, $(3+1)$ ghost odd-valued fields $C^p$, $\widehat{B}$ and $3$ even-valued $B^{pq}$ for $p$, $q=1,2,3$. To construct the quantum action, $S_{(3)}$, by adding to the classical action, $S_0({𝒜})$, of an $N=3$-exact gauge-fixing term (with gauge fermion), a gauge-fixing procedure requires $(1+3+3+1)$ additional fields, ${\bar{\mathrm{\Phi }}}_{(3)}$: antighost $\bar{C}$, $3$ even-valued $B^p$, 3 odd-valued ${\widehat{B}}^{pq}$ and Nakanishi–Lautrup $B$ fields. The action of $N=3$ transformations on new fields as $N=3$-irreducible representation space is realized. These transformations are the $N=3$ BRST symmetry transformations for the vacuum functional, $Z_3(0)=\int d{\mathrm{\Phi }}_{(3)}d{\bar{\mathrm{\Phi }}}_{(3)}exp\{(ı/\hslash )S_{(3)}\}$. The space of all fields $({\mathrm{\Phi }}_{(3)},{\bar{\mathrm{\Phi }}}_{(3)})$ proves to be the space of an irreducible representation of the fields ${\mathrm{\Phi }}_{(4)}$ for $N=4$-parametric SUSY transformations, which contains, in addition to ${{𝒜}}^{\mu }$ the $(4+6+4+1)$ ghost–antighost, $C^r=(C^p,\bar{C})$, even-valued, $B^{rs}=-B^{sr}=(B^{pq},B^{p4}=B^p)$, odd-valued ${\widehat{B}}^r=(\widehat{B},{\widehat{B}}^{pq})$ and $B$ fields. The quantum action is constructed by adding to $S_0({𝒜})$ an $N=4$-exact gauge-fixing term with a gauge boson, $F_{(4)}$. The $N=4$ SUSY transformations are by $N=4$ BRST transformations for the vacuum functional, $Z_4(0)=\int d{\mathrm{\Phi }}_{(4)}exp\{(ı/\hslash )S_{(4)}\}$. The procedures are valid for any admissible gauge. The equivalence with $N=1$ BRST-invariant quantization method is explicitly found. The finite $N=3,4$ BRST transformations are derived and the Jacobians for a change of variables related to them but with field-dependent parameters in the respective path integral are calculated. They imply the presence of a corresponding modified Ward identity related to a new form of the standard Ward identities and describe the problem of a gauge-dependence. An introduction into diagrammatic Feynman techniques for $N=3,4$ BRST invariant quantum actions for Yang–Mills theory is suggested.

To reduce general relativity to the canonical Hamiltonian formalism and construct the path (functional) integral in a simpler and, especially in the discrete case, less singular way, one extends the configuration superspace, as in the connection representation. Then we perform functional integration over connection. The module of the result of this integration arises in the leading order of the expansion over a scale of the discrete lapse-shift functions and has maxima at finite (Planck scale) areas/lengths and rapidly decreases at large areas/lengths, as we have mainly considered previously; the phase arises in the leading order (Regge action) of the stationary phase expansion.

Now we consider the possibility of confining ourselves to these leading terms in a certain region of the parameters of the theory; consider background edge lengths as an optimal starting point for the perturbative expansion of the theory; estimate the background length scale and consider the form of the graviton propagator. In parallel with the general simplicial structure, we consider the simplest periodic simplicial structure with a part of the variables frozen (“hypercubic”), for which also the propagator in the leading approximation over metric variations can be written in a closed form.

This paper is in the spirit of our work on the consequences of the Regge calculus, where some edge length scale arises as an optimal initial point of the perturbative expansion after functional integration over connection. Now consider the perturbative expansion itself. To obtain an algorithmizable diagram technique, we consider the simplest periodic simplicial structure with a frozen part of the variables (“hypercubic”). After functional integration over connection, the system is described by the metric $g_{\lambda \mu }$ at the sites. We parameterize $g_{\lambda \mu }$ so that the functional measure becomes Lebesgue. The discrete diagrams are free from ultraviolet divergences and reproduce (for ordinary, non-Planck external momenta) those continuum counterparts that are finite. We give the parametrization of $g_{\lambda \mu }$ up to terms, providing, in particular, additional three-graviton and two-graviton-two-matter vertices, which can give additional one-loop corrections to the Newtonian potential. The edge length scale is $\sim \sqrt{\eta }$, where $\eta$ defines the free factor ${(-det\parallel g_{\lambda \mu }\parallel )}^{\eta ∕2}$ in the measure and should be a large parameter to ensure the true action after integration over connection. We verify the important fact that the perturbative expansion does not contain increasing powers of $\eta$ if its initial point is chosen close enough to the maximum point of the measure, thus justifying this choice. Discrete propagators depend on the Barbero–Immirzi parameter $\gamma$, which determines the ratio of timelike and spacelike elementary length scales. The existing estimates of $\gamma$ allow the propagator poles to have real energy for any (real) spatial momenta.

An attempt to directly use the synchronous gauge ($g_{0\lambda }=-{\delta }_{0\lambda }$) in perturbative gravity leads to a singularity at $p_0=0$ in the graviton propagator. This is similar to the singularity in the propagator for Yang–Mills fields $A_{\lambda }^a$ in the temporal gauge ($A_0^a=0$). There the singularity was softened, obtaining this gauge as the limit at $\epsilon \to 0$ of the gauge $n^{\lambda }{A}_{\lambda }^a=0$, $n^{\lambda }=(1,-\epsilon {({\partial }^j{\partial }_j)}^{-1}{\partial }^k)$. Then the singularities at $p_0=0$ are replaced by negative powers of $p_0\pm i\epsilon$, and thus we bypass these poles in a certain way. Now consider a similar condition on $n^{\lambda }{g}_{\lambda \mu }$ in perturbative gravity, which becomes the synchronous gauge at $\epsilon \to 0$. Unlike the Yang–Mills case, the contribution of the Faddeev–Popov ghosts to the effective action is nonzero, and we calculate it. In this calculation, an intermediate regularization is needed, and we assume the discrete structure of the theory at short distances for that. The effect of this contribution is to change the functional integral measure or, for example, to add nonpole terms to the propagator. This contribution vanishes at $\epsilon \to 0$. Thus, we effectively have the synchronous gauge with the resolved singularities at $p_0=0$, where only the physical components $g_{jk}$ are active and there is no need to calculate the ghost contribution.

In this paper, we computed quantum friction of two parallel metal plates separated by a small distance moving with constant relative velocity $v=\left|v\right|$. The plasmons as the internal degrees of freedom living on the two plates are coupled to a vacuum field in the gap between the two plates. We got the in–out quantum action which contained all the dynamical information of the system. Furthermore, we associated the imaginary part of the in–out quantum action with dissipation and frictional force. For the case of dispersionless plasmons, the imaginary part of the in–out quantum action is strongly suppressed as $v\to 0$. The frictional force exhibits the same feature as $v\to 0$. The difference is that the frictional force increases as $v<0.015$ and decreases as $v>0.015$. For the case of dispersive plasmons, there is a threshold for the imaginary part of the in–out quantum action and the frictional force, that is, there is no dissipation when the relative velocity $v$ is not big enough. We gave a classical argument of the existence of the threshold, and this argument matched the mathematical results.

This paper is an introduction to combinatorial knot theory via state summation models for the Jones polynomial and its generalizations. It is also a story about the developments that ensued in relation to the discovery of the Jones polynomial and a remembrance of Vaughan Jones and his mathematics.

This paper focuses on the average values of functionals like $Y={\int }_0^{1}g(x(t))dt$ on the set $M=\{x|\parallel x{\parallel }_p\le R,x\in C[0,1]\}$ in $L^p[0,1]$. The densities of coordinates of points in M are derived out. The formula of average value EY of functional Y is obtained. The variance DY of Y is proven to be zero, which shows the phenomenon of concentration without measure, and then the nonlinear commutation identity Eh(Y) = h(EY) is obtained for continuous function $h$. Finally, particularly, it is proven that the average value depends on the discretization.

This book provides a concise introduction to quantum mechanics for students versed in mathematics and mechanics. Dr Strocchi imparts a holistic perspective of the quantum landscape by focussing on three concepts: the algebra of observables, basic quantum systems, and stochastic processes. After mastering these concepts, readers are prepared to study specific quantum theories of information, statistical mechanics, and fields.

The basic principles of Affine Quantum Gravity are presented in a pedagogical style with a limited number of equations.

In the space-of-histories approach to gauge fields and their quantization, the Maxwell, Yang–Mills and gravitational field are well known to share the property of being type-I theories, i.e. Lie brackets of the vector fields which leave the action functional invariant are linear combinations of such vector fields, with coefficients of linear combination given by structure constants. The corresponding gauge-field operator in the functional integral for the in-out amplitude is an invertible second-order differential operator. For such an operator, we consider advanced and retarded Green functions giving rise to a Peierls bracket among group-invariant functionals. Our Peierls bracket is a Poisson bracket on the space of all group-invariant functionals in two cases only: either the gauge-fixing is arbitrary but the gauge fields lie on the dynamical sub-space; or the gauge-fixing is a linear functional of gauge fields, which are generic points of the space of histories. In both cases, the resulting Peierls bracket is proved to be gauge-invariant by exploiting the manifestly covariant formalism. Moreover, on quantization, a gauge-invariant Moyal bracket is defined that reduces to iħ times the Peierls bracket to lowest order in ħ.

In the case of gauge theories, which are ruled by an infinite-dimensional invariance group, various choices of antisymmetric bilinear maps on field functionals are indeed available. This paper proves first that, within this broad framework, the Peierls map (not yet the bracket) is a member of a larger family. At that stage, restriction to gauge-invariant functionals of the fields, with the associated Ward identities and geometric structure of the space of histories, make it possible to prove that the new map is indeed a Poisson bracket in the simple but relevant case of Maxwell theory. The building blocks are available for gauge theories only: vector fields that leave the action functional invariant; the invertible gauge-field operator; and the Green function of the ghost operator.

After an overview of the physical motivations for studying quantum gravity, we reprint THE FORMAL STRUCTURE OF QUANTUM GRAVITY, i.e. the 1978 Cargèse Lectures in Levy & Deser (1979): Recent Developments in Gravitation pp. 275–322 by Professor B. S. DeWitt, with kind permission of Springer Science and Business Media. The reader is therefore introduced, in a pedagogical way, to the functional integral quantization of gravitation and Yang–Mills theory. It is hoped that such a paper will remain useful for all lecturers or Ph.D. students who face the task of introducing (resp. learning) some basic concepts in quantum gravity in a relatively short time. In the second part, we outline selected topics such as the braneworld picture with the same covariant formalism of the first part, and spectral asymptotics of Euclidean quantum gravity with diffeomorphism-invariant boundary conditions. The latter might have implications for singularity avoidance in quantum cosmology.

The functional integral for the quantization of the coadjoint orbits of the unitary and orthogonal groups is given by means of an explicit construction of the corresponding «Darbouxi» variables.

A sketch of a recent approach to quantum gravity is presented which involves several unconventional aspects. The basic ingredients include: (1) Affine kinematical variables; (2) Affine coherent states; (3) Projection operator approach for quantum constraints; (4) Continuous-time regularized functional integral representation without/with constraints; and (5) Hard core picture of nonrenormalizability. Emphasis is given to the functional integral expressions.

The appearance of coherence and order is studied for ultracold bosonic atoms in the presence of additional disorder potentials. These arise either naturally like in current carrying wire traps, or artificially and controllably like in laser speckle fields. The description of such disordered bosons within a suitably generalized Bogoliubov theory, first given by Huang and Meng, is rederived here within a functional integral approach for replicated bosonic fields. The superfluidity in homogeneous Bose systems with condensates depleted by weak interactions and disorder can thereby be discussed.

A mixture of spin-1/2 fermionic atoms and molecules of paired fermionic atoms is studied in an optical lattice. The molecules are formed by an attractive nearest-neighbor interaction. A functional integral is constructed for this many-body system and analyzed in terms of a mean-field approximation and Gaussian fluctuations. This provides a phase diagram with the two merging Mott insulators and an intermediate superfluid. The Gaussian fluctuations give rise to an induced repulsive dimer-dimer interaction mediated by the unpaired fermions. The effect of an unbalanced distribution of spin-up and spin-down fermions is also discussed.

A mixture of two types of fermions with different mass in an optical lattice is considered. The light fermions are subject to thermal and quantum fluctuations, the heavy fermions only to thermal fluctuations. We derive the distribution of heavy fermions and study numerically the localization length of the light fermions in a two-dimensional lattice. Depending on the temperature of the system, a transition from extended states at low temperatures to localized states at high temperatures is found.

The dilute Bose gas is studied in the large-N limit using functional integration.