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Path integral
Jean Zinn-Justin (2009), Scholarpedia, 4(2):8674. | doi:10.4249/scholarpedia.8674 | revision #147600 [link to/cite this article] |
A sizable fraction of the theoretical developments in physics of the last sixty years would not be understandable without the use of path or, more generally, field integrals. In this article we will focus on the use of path integrals and field integrals in different branches of theoretical physics. A rigorous study of the mathematical properties of path and field integrals is an open subtopic of functional analysis and will not be dealt with here.
Path integrals are given by sum over all paths satisfying some boundary conditions and can be understood as extensions to an infinite number of integration variables of usual multi-dimensional integrals. Path integrals are powerful tools for the study of quantum mechanics. Indeed, in quantum mechanics, physical quantities can be expressed as averages over all possible paths weighted by the exponential of a term proportional to the ratio of the classical action \(\mathcal S \) associated to each path, divided by the Planck's constant \( \hbar\ .\) Thus, path integrals emphasize very explicitly the correspondence between classical and quantum mechanics. In particular, in the semi-classical limit \(\mathcal{S}/\hbar\rightarrow\infty\ ,\) the leading contributions in the average come from paths close to classical paths, which are stationary points of the action. Thus, path integrals lead to an intuitive understanding and simple calculations of physical quantities in the semi-classical limit.
The formulation of quantum mechanics based on path integrals is well adapted to systems with many degrees of freedom, where a formalism of Schrödinger type is much less useful. Therefore, it allows an easy transition from quantum mechanics to quantum field theory or statistical physics. In particular, generalized path integrals (functional integrals and, more precisely, field integrals) lead to an understanding of the deep relations between quantum field theory and the theory of critical phenomena in continuous phase transitions.
We first describe Brownian motion and Euclidean-time (i.e, imaginary time) path integrals. This means that we consider the path integral representation of the matrix elements of the quantum statistical operator, or density matrix at thermal equilibrium \(\mathrm{e}^{-\beta\hat H}, \) \(\hat H\) being the quantum Hamiltonian and \(\beta\) the inverse temperature (measured in a unit where the Boltzmann constant \(k_B\) is 1). In this way, we are able to describe quantum statistical physics in terms of path integrals, but also, perhaps more surprisingly, to exhibit a relation between classical and quantum statistical mechanics. Moreover, for a whole class of Hamiltonians, the Euclidean-time path integral corresponds to a positive measure. We then define the real-time (in relativistic field theory Minkowskian-time) path integral, which describes the time evolution of quantum systems and corresponds for time-translation invariant systems to the evolution operator \(\mathrm{e}^{-i t\hat H/\hbar}\) (\(t\) being the real time). Finally, we briefly list a few generalizations: path integrals in the Hamiltonian formulation, path integrals in the holomorphic representation related to boson systems and, correspondingly, Grassmannian path integrals for fermions.
A number of important applications to physics of the path integral idea involve in fact integrals over fields. In particular, field integrals are indispensable for the study of quantum gauge invariant theories which constitute the basis of the description of fundamental interactions at the microscopic scale, as well as for understanding of the critical properties of phase transitions. They rely on a pragmatic approach, focusing more on developing calculational tools than on establishing rigorous properties. Indeed, even though a number of interesting rigorous results have been proved, one faces extremely difficult mathematical problems in realistic situations (e.g., in four dimensional space-time).
Random walk, Brownian motion and path integral
As a first example, we consider a random walk on the real line with discrete times \(k=0,1,2,\dots, n.\) Such a stochastic process is specified by a probability distribution \(P_0(x)\) for the position \(x\) at initial time \( k=0\) and a time-independent density \(\rho(x, x')\) describing the probability of transition from the point \( x'\) to the point \(x\ ,\) meaning that the probability distribution \(P_k(x)\) at time \(k\) satisfies the recursion relation or master equation \[P_k(x)=\int\mathrm{d}x'\,\rho(x,x')P_{k-1}(x'), \quad \int\mathrm{d} x\,\rho(x,x')=1\,.\] This random walk is a (homogeneous or stationary) Markov chain, that is, a (time-translation invariant) stochastic process with the Markov property (i.e., the property that the probability distribution at time \(k\) depends only on the probability at time \(k-1\ ,\) but not on the probability distributions at prior times).
In the following we also assume that the transition probability is translation invariant and even\[\rho(x, x')=R(x-x')=R(x'-x)\ .\] Under rather general conditions, the most important being that \(R(x)\) decreases fast enough for \(|x|\) large (we call this a local Markov process), one can prove (a consequence of the central limit theorem of probabilities) that the distribution \(P_n(x)\) converges asymptotically for large times \(n\) toward a Gaussian distribution that is independent of the transition probability \(R(x)\ .\) Therefore, if one is interested only in large time properties, one can start directly from a Gaussian transition probability of the form \[\tag{1} R(x)={1\over\sqrt{2\pi\xi}}\mathrm{e}^{-x^2/(2\xi)},\]
where \(\xi>0\) characterizes the width of the distribution. It is then easy to calculate \(P_n(x)\) explicitly by successive Gaussian integrations. However, for our purpose it is more instructive to just apply the recursion relation. If one assumes, for example, that the initial distribution is concentrated at the point \(x=x_0\) (i.e., \(P_0(x)=\delta(x-x_0)\) where \(\delta(x)\) is Dirac's generalized function also known as Dirac function or \(\delta-\)function) one obtains at time \(n\) the probability distribution \[\tag{2} P_n(x,x_0)=\int\mathrm{d} x_{n-1}\mathrm{d} x_{n-2}\ldots \mathrm{d} x_{1}\,R(x-x_{n-1})R(x_{n-1}-x_{n-2})\ldots R(x_1-x_0).\]
In the Gaussian example (1), the expression becomes \[\tag{3} P_n(x,x_0)=(2\pi\xi)^{(1-n)/2}\int\mathrm{d} x_{n-1}\mathrm{d} x_{n-2}\ldots \mathrm{d} x_{1}\,\mathrm{e}^{-\mathcal{S}(\mathbf{x})/\xi}\], where, defining \(\mathbf{x}\equiv(x_0,x_1,\cdots,x_n)\) and \(x\equiv x_n\,\) \[\mathcal{S}(\mathbf{x})={1\over2}\sum_{k=1}^{n}\left(x_k-x_{k-1}\right)^2\,\].
We then introduce a time step \(\varepsilon>0\ ,\) the macroscopic time variables \[\tau_k=t'+k\varepsilon\ \mathrm{with}\ 0\le k\le n\,,\] (such that \(\tau_0=t'\ ,\) \(\tau_n=t'+n\varepsilon\equiv t''\)) and a continuous, piecewise linear path (see Figure 1) \[\tag{4} q(\tau)=\sqrt{\varepsilon}\left[ x_{k-1}+{\tau-\tau_{k-1} \over \tau_k-\tau_{k-1}}\left( x_k-x_{k-1}\right)\right]\quad \mathrm{for}\ \tau_{k-1}\le \tau\le \tau_k\quad \mathrm{and}\; k\ge 1,\]
with the boundary conditions \[\tag{5} q(t')=\sqrt{\varepsilon}x_0\equiv q'\,,\quad q(t'')=\sqrt{\varepsilon} x\equiv q''\,.\]
One verifies that \(\mathcal{S}(\mathbf{x}) \) can be rewritten as \[\mathcal{S}(\mathbf{x})=\mathcal{S}_\varepsilon(\mathbf{q})\equiv {1\over 2}\int_{t'}^{t''} \,\dot {q}^2(\tau)\mathrm{d} \tau\] where \(\dot{q}(\tau)\equiv \mathrm{d}q/\mathrm{d}\tau\ .\)
One can then study the large discrete time asymptotic behaviour, by taking the large \(n\) limit at \(t''-t'\) fixed and, thus, \(\varepsilon=(t''-t')/n\to 0\ .\) One also speaks of a temporal continuum limit since the time step goes to zero. In this limit, the normalized probability distribution in the new variables \(\Pi_0(t'',t';q'',q')\) is given by a Euclidean-time path integral (Wiener 1923) that we denote by \[\tag{6} \Pi_0(t'',t';q'',q')=\lim_{n\to\infty}{1\over\sqrt{\varepsilon}}P_n(x,x_0)=\int[\mathrm{d} q(\tau)]\mathrm{e}^{-\mathcal{ S}_0(\mathbf{q})/\xi},\quad \mathcal{S}_0(\mathbf{q})={1\over 2}\int_{t'}^{t''} \,\dot{q}^2(\tau)\mathrm{d} \tau\]
(the factor \(1/\sqrt{\varepsilon}\) comes from the change of variables from \(x\) to \(q\)) where the symbol \([\mathrm{d} q(\tau)]\) (also denoted by \(\mathcal{D}q(\tau)\) in the literature) means sum over all (trajectories) \(q(\tau)\) satisfying the boundary conditions (5).
A few simple remarks are in order. First, the integrand in the path integral is positive and \( [\mathrm{d} q(\tau)]\mathrm{e}^{-\mathcal{ S}_0(\mathbf{q})/\xi}\) thus defines a positive measure on paths, the so-called Wiener measure. Second, it is difficult to keep track of the absolute normalization in the continuum path integral limit. Therefore, one mostly uses path integrals to calculate expectation values. If \(\mathcal{F}(\mathbf{q})\) is a functional of the path \(\mathbf{q}\equiv q(\cdot),\) its expectation value is defined by \[\tag{7} \langle\mathcal{F}(\mathbf{q})\rangle_0 =\mathcal{Z}^{-1}\int[\mathrm{d} q(\tau)]\mathcal{F}(\mathbf{q})\mathrm{e}^{-\mathcal{ S}_0(\mathbf{q})/\xi}\]
with \[\mathcal {Z}\equiv \int[\mathrm{d} q(\tau)] \mathrm{e}^{-\mathcal{ S}_0(\mathbf{q})/\xi}.\] Note that in the ratio of (7) the normalization cancels. In the case of the random walk, typical expectation values correspond to correlations between positions at different times, which can also be considered as generalized moments of the probability distribution. For example, \[\mathcal{F}(\mathbf{q})\equiv q(\tau_1)q(\tau_2)\ldots q(\tau_{2n}).\] Expectation values of this kind are also called correlation functions.
The form of \(\mathcal{ S}_0(\mathbf{q})\) determines the class of paths that contribute to the path integral, which are called in this case Brownian paths. As the factor \(\sqrt{\varepsilon}\) in (4) suggests, Brownian paths satisfy a Hölder condition of order 1/2, that is, for \(\tau-\tau'\to0\ ,\) \[ | q(\tau)-q(\tau') | = O \left( \left| \tau-\tau' \right|^{1/2} \right);\] in particular Brownian paths are continuous but not differentiable and, thus, \(\dot{q}(\tau)\) is not defined. In this sense, the notation \( \dot{q}^2(\tau)\) has to be considered as a symbol and should not taken literally. Nevertheless, it is a useful notation since, for \(\xi\to0\ ,\) the path integral is dominated by paths close to the classical paths that leave \(\mathcal{ S}_0(\mathbf{q})\) stationary, and which are differentiable (see the calculation below). Finally, the continuity of the paths allows understanding why it is possible that an integration over an increasingly dense set of points (or sum over all possible piecewise linear paths) eventually converges toward an integration over all continuous paths.
Explicit calculation
Gaussian path integrals, like finite dimensional Gaussian integrals, are explicitly calculable. Let us illustrate this property with the simple example of the Brownian motion. We now explicitly evaluate the integral (6) by a method that generalizes to other kinds of Gaussian integrals. Varying the quantity \(S_0(\mathbf{q})\) with respect to the path \(q(\tau),\) one obtains the classical equation of motion \(\ddot q(\tau)=0\ .\) Imposing the boundary conditions (5) of the path integral to the classical solution, one obtains \[q_c(\tau)=q'+\frac{(\tau-t')}{(t''-t')}(q''-q').\] One then changes variables, \(q(\tau)\mapsto r(\tau)\) with \(q(\tau)=q_c(\tau)+r(\tau).\) At each time \(\tau\) this is a simple translation and the associated Jacobian is \(1\ .\) The boundary conditions on the new path \( r(\tau)\) are \(r(t'')=r(t')=0\) and one finds \[S_0(\mathbf{q})=\frac{1}{2}{(q''-q')^2\over(t''-t')}+S_0(\mathbf{r}).\] The path integral (6) becomes \[\Pi_0(t'',t';q'',q')=\mathrm{e}^{-\frac{1}{2\xi}{(q''-q')^2\over(t''-t')}} \int[\mathrm{d} r(\tau)]\mathrm{e}^{-\mathcal{ S}_0(\mathbf{r})/\xi}.\] The remaining path integral is \(q'', q'\)-independent and gives then a simple normalization factor. Here, it can be determined by imposing the condition of probability conservation \[\tag{8} \int\mathrm{d}q''\,\Pi_0(t'',t'; q'',q')=1\ \Rightarrow\ \Pi_0(t'',t';q'',q')={1\over\sqrt{2\pi\xi(t''-t')}}\mathrm{e}^{-\frac{(q''-q')^2}{2(t''-t')\xi}}.\]
As pointed out above, this normalization cancels in expectation values.
Application of the Wiener measure to statistical physics
The same path integral describing the Brownian motion has an interpretation in the framework of statistical physics.
Classical statistical physics
The expression (3) in the example (1) may also be physically interpreted as the classical partition function of \(n+1\) particles on a one-dimensional lattice with spatial sites \(k=0,1,\cdots, n\ .\) Particles deviate from their equilibrium positions by the value \(x_k\) and have nearest-neighbour harmonic interactions: \[\mathcal {Z}(x_n,x_0;\xi)=(2\pi\xi)^{(1-n)/2}\;\int\mathrm{d} x_{n-1}\mathrm{d} x_{n-2}\ldots \mathrm{d} x_{1} \,\exp\left(-{1\over2\xi}\sum_{k=1}^{n}\left(x_k-x_{k-1}\right)^2\right).\] (The extremities of the chain being fixed at deviations \(x_0\) and \(x_n\ ,\) respectively.) Here, the parameter \(\xi\) has the interpretation of a temperature. The path integral (6) then corresponds to the continuum limit where the lattice spacing \(\varepsilon \) between two adjacent sites goes to zero at fixed total macroscopic length of the chain \(L=n\varepsilon\ .\)
Quantum statistical physics
Remarkably enough, the path integral of the Brownian motion yields also the density matrix of a free non-relativistic quantum particle.
The continuum distribution \(\Pi_0(t,t';q,q')\) given by equation (8) satisfies the diffusion equation \[{\partial \Pi_0 \over\partial t}={\xi \over2}{\partial^2 \Pi_0 \over (\partial q)^2} \;,\] with initial condition: \[\lim_{t\rightarrow t'}\Pi_0(t,t';q,q')=\delta(q-q')\;.\] This equation can be compared with the equation satisfied by the elements of the quantum density matrix \(\mathrm{e}^{-\beta \hat{H}_0}\) (in the basis in which the position operator \(\hat q\) is diagonal \(:\;\hat {q}|q\rangle= q |q\rangle\)), an operator describing the thermal equilibrium of a free non-relativistic quantum particle, where \(\hat{H}_0={\hat{p}}^2/(2 m)\) is the quantum Hamiltonian in real time (a linear operator acting on the Hilbert space of quantum states), \(\hat p\) the momentum operator (with the commutation relation \([\hat q,\hat p]=i\hbar \)), \(m\) is the particle's mass and \(\beta\) the inverse temperature (in units of the Boltzmann's constant \(k_\mathrm{B}\)). The matrix elements \(\langle q |\mathrm{e}^{-\beta \hat{H}_0}|q'\rangle\) satisfy the partial differential equation: \[{\partial \over \partial \beta} \langle q | \mathrm{e}^{-\beta \hat{H}_0}|q'\rangle ={\hbar^2\over 2m}{\partial^2 \over (\partial q)^2}\langle q |\mathrm{e}^{-\beta \hat{H}_0}|q'\rangle,\] \( \hbar\) being Planck's constant. Since \(\Pi_0(t,t';q,q')\) and \(\langle q | \mathrm{e}^{-\beta \hat{H}_0}|q'\rangle\) also satisfy the same boundary conditions at initial time \(t-t'\equiv\hbar\beta=0\ ,\) it follows that \(\langle q |\mathrm{e}^{-\beta \hat{H}_0}|q'\rangle\) is equal to \(\Pi_0(t=\hbar\beta,0;q,q')\) when \(\xi=\hbar/m\) and, therefore, is given by the path integral
\[\tag{9} \langle q |\mathrm{e}^{-\beta \hat{H}_0}|q'\rangle =\int[\mathrm{d} q(\tau)]\mathrm{e}^{-\mathcal{ S}_0(\mathbf{q})/\hbar},\quad \mathcal{S}_0(\mathbf{q})={1\over 2}\int_{0}^{\hbar\beta} \,m\dot{q}^2(\tau)\mathrm{d} \tau \]
with the boundary conditions \(q(0)=q',\) \(q(\hbar\beta)=q.\)
A simple generalization of the path integral (9) relevant for quantum statistical physics is the path integral \[\tag{10} \Pi(t'',t';q'',q')=\int[\mathrm{d}q(\tau)]\mathrm{e}^{-\mathcal{S}(\mathbf{q})/\hbar} \quad\text{with}\; q(t')=q', q(t'')=q''\;,\]
where \[\tag{11} \mathcal{S}(\mathbf{q})=\int_{t'}^{t'' }\mathrm{d}\tau\,\mathcal{L}_{\mathrm{E}}(\dot {q}(\tau),q(\tau);\tau)\]
and the Euclidean Lagrangian is defined as \[\tag{12} \mathcal {L}_{\mathrm{E}}(\dot{q},q;\tau)= \frac{1}{2}m\dot{q}^2+V(q,\tau)\;.\]
Note that in the Euclidean Lagrangian the potential \(V(q,\tau)\) is added to the kinetic energy, while in the normal Lagrangian of classical mechanics the potential is subtracted from the kinetic energy. The parameter \(m\) can be identified with the mass of a non-relativistic quantum particle. We also assume that \(V(q,t)\) is a smooth function of \(q\) and that \[\tag{13} \int\mathrm{d}q\, \mathrm{e}^{-\varepsilon V(q,\tau)}<\infty\ \forall \tau \ \mathrm{and}\ \forall \varepsilon>0\,.\]
One possible definition of this kind of path integrals refers to the Wiener measure: \[\tag{14} \Pi(t'',t';q'',q')=\Pi_0(t'',t';q'',q')\left\langle \exp\left[-{1\over\hbar}\int_{t'}^{t'' }\mathrm{d}\tau\, V(q(\tau),\tau)\right]\right\rangle_0\,,\]
where the expectation value is defined in (7) with \(\xi=\hbar/m.\) With this normalization, \(\Pi(t',t';q'',q')=\delta(q''-q')\ ,\) which is the kernel associated with the identity operator.
Path integrals and local Markov processes
We introduce inside the path integral (10) the identity \[1=\int\mathrm{d} q\,\delta\bigl(q-q(t)\bigr) \ \mathrm{with}\ t'<t<t'',\] where \(\delta(q)\) is Dirac's \(\delta\)-function. Also, the expression (11) can be written as the sum \[ \mathcal{S}(\mathbf{q})=\int_{t'}^t\mathrm{d}\tau\,\mathcal {L}_{\mathrm{E}}(\dot{q},q;\tau) +\int_{t}^{t''}\mathrm{d}\tau\,\mathcal {L}_{\mathrm{E}}(\dot{q},q;\tau).\] The path integral thus factorizes into the product of two path integrals with boundary conditions at \( t''\) and \( t\ ,\) and \( t\) and \( t'\ ,\) respectively, integrated over the intermediate point \(q\ .\) One concludes that \(\Pi(t'',t';q'',q')\) defined by the path integral (10) satisfies a Markov property in (Euclidean) time: \[\tag{15} \Pi(t'',t';q'',q')=\int\mathrm{d}q\; \Pi(t'',t;q'',q)\; \Pi(t,t';q,q')\quad \mathrm{for}\ t''\ge t\ge t' \,. \]
This multiplication rule also shows that \(\Pi(t'',t';q'',q')\) can be identified with the kernel or matrix element of an operator \(\boldsymbol{\Pi}(t'',t')\) in an Hilbert space \( \mathcal{H}\ .\) In Dirac's bra-ket notation, \[\tag{16} \langle q''|\boldsymbol{\Pi}(t'',t')|q'\rangle\equiv \Pi(t'',t';q'',q').\]
In operator notation, the relation (15) becomes \[\boldsymbol{\Pi}(t'',t')=\boldsymbol{\Pi}(t'',t)\boldsymbol{\Pi}(t,t').\] The Markov property (15) allows writing the path integral as a product of \(n\) path integrals integrated over intermediate points, corresponding to a time interval \(\varepsilon=(t''-t')/n\) that can be chosen arbitrarily small by increasing \(n\ .\) The path integral can thus be evaluated from its asymptotic behaviour for small time intervals. Then, with the boundary conditions \(q(t)=q,\) \(q(t+\varepsilon)=q_\varepsilon\ ,\) \[\tag{17} \mathcal{S}(\mathbf{q})=\int_{t}^{t+\varepsilon}\mathrm{d}\tau\left[\frac{1}{2}m\dot{q}^2(\tau)+V(q(\tau),\tau)\right]\sim m{(q_\varepsilon-q)^2\over2\varepsilon}+\varepsilon V(q,t). \]
For a given trajectory, the leading term in the expression when \(\varepsilon\rightarrow0\) is still the Brownian term (kinetic term in classical mechanics), which implies that the paths contributing to the path integral still satisfy the Hölder property \(|q_\varepsilon-q|=O(\varepsilon^{1/2})\ .\) In particular, this property implies that the argument \(q\) of \(V\) in (17) can be replaced by any value linearly interpolating between \(q\) and \(q_\varepsilon\ ,\) the difference being of order \( \varepsilon^{1/2}\) hence negligible in this approximation. The dominance of the Brownian term implies also a (spatial) locality property (and thus the denomination of local Markov process): for \(\varepsilon\) small, \(\langle q_\varepsilon|\boldsymbol{\Pi}(t+\varepsilon,t)|q\rangle\) decreases exponentially when \(|q-q_\varepsilon|\to \infty\ .\) Since at leading order in \(\varepsilon\) the normalization is provided by the Brownian motion, one concludes that \[\tag{18} \langle q_\varepsilon|\boldsymbol{\Pi}(t+\varepsilon,t)|q\rangle\sim \Pi_\varepsilon(t;q_\varepsilon,q)=\sqrt{m\over2\pi\hbar\varepsilon} \exp\left[-\left(m{(q_\varepsilon-q)^2\over2\varepsilon}+\varepsilon V(q,t)\right)/\hbar\right] .\]
This leads to an alternative definition of the path integral (10) as the limit when \(n\to\infty\ \) at \(n\varepsilon=t''-t'\) fixed of an \(n\)-dimensional integral: \[\tag{19} \langle q'' | \boldsymbol{\Pi} (t'' ,t' ) | q' \rangle =\lim_{n\to \infty} \int \prod ^{n-1}_{k=1}\mathrm{d} q_{k} \prod^{n}_{k=1} \Pi_\varepsilon(\tau_{k-1};q_k,q_{k-1})\]
with the conventions \(\tau_{k}=t' +k\varepsilon\ ,\) q\(_{0}=q'\ ,\) \(q_{n}=q ''\ .\) At finite \(n\) the right hand side defines a Markov process in discrete times.
Path integrals and classical statistical physics
If the interpretation of the path integral in terms of random walks (see (14)) and Markov processes is somewhat indirect, the interpretation in the framework of classical statistical physics is simple. In the form of the right hand side of (19), the path integral appears as the continuum limit of the classical partition function of a one-dimensional lattice model. The configuration energy of the lattice model is (here we set \(m=\hbar=1\)) \[S(\mathbf{q})=\sum_{k=1}^{n}\left[{1\over2\varepsilon}\left(q_k-q_{k-1}\right)^2+\varepsilon V(q_k)\right]\,,\] where we have assumed \(V\) time independent to enforce translation invariance. The continuum limit \(\varepsilon\propto 1/ n \to 0\) corresponds also to a kind of low temperature limit where the correlation length in the statistical model, which is proportional to \(1/\varepsilon\ ,\) diverges.
Path integrals and quantum statistical physics
From the evaluation of the path integral at short time intervals, one can also infer that \(\langle q'' | \boldsymbol{\Pi} (t'' ,t' ) | q'\rangle\) satisfies the partial differential equation \[\tag{20} \hbar {\partial \over \partial t}\langle q | \boldsymbol{\Pi} (t ,t' ) | q'\rangle=\left[{\hbar^2\over 2m}{\partial^2\over (\partial q)^2}-V(q,t)\right] \langle q | \boldsymbol{\Pi} (t ,t' ) | q'\rangle\;,\]
with the initial condition\( \langle q | \boldsymbol{\Pi}(t' ,t') | q'\rangle =\delta (q-q')\) (i.e., \(\boldsymbol{\Pi} (t',t' ) = \mathbf{1}\)).
When the potential is time-independent, that is, when \(V(q,t)=V(q)\ ,\) equation (20) is related by the formal substitution \(t\mapsto it\) to the Schrödinger equation in real time: \[\tag{21} i\hbar{\partial \over \partial t }\langle q | \mathbf{U} (t ,t' ) | q'\rangle=\left[-{\hbar^2\over 2m}{\partial^2\over (\partial q)^2}+V(q)\right] \langle q | \mathbf{U} (t ,t' ) | q'\rangle\,, \]
for the matrix elements of the quantum evolution operator \(\mathbf{U} (t'' ,t' ).\) The solution of (21) with initial condition \(\mathbf{U}(t' ,t' )=\mathbf{1}\) is \[\tag{22} \mathbf{U}(t ,t')= \mathrm{e}^{-i (t -t' )\hat{H}/\hbar} \;, \]
where \[\tag{23} \hat{H}=\frac{{\hat p}^2}{2 m}+V(\hat{q})\;, \]
is a time-independent quantum Hamiltonian and \(\hat q\) and \(\hat p\) are, respectively, the position and momentum operators. Note also that with the condition (13), the Hamiltonian (23) has a discrete spectrum.
Equation (20), also called imaginary time Schrödinger equation, has then a solution \(\boldsymbol{\Pi}(t,t') =\mathrm{e}^{- (t -t' )\hat{H}/\hbar}\) with initial condition \(\boldsymbol{\Pi}(t,t)=\mathbf{1}\) and, therefore, \[\tag{24} \boldsymbol{\Pi}(t=\beta\hbar,0)= \mathrm{e}^{-\beta \hat{H}}\,, \]
where \(\mathrm{e}^{-\beta \hat{H}}\) is the density matrix operator describing the thermodynamic equilibrium of a quantum system with quantum Hamiltonian \(\hat{H}\) at temperature \( {1}/{\beta}\ .\) Equations (10), (11), (16) and (24) generalize (9).
The partition function \(\mathcal{Z}(\beta)=\mathrm{tr} \,\mathrm{e}^{-\beta\hat H}=\mathrm{tr}\boldsymbol{\Pi}(\hbar\beta/2,-\hbar\beta/2)\) of the quantum system can be represented as \[\tag{25} \mathcal{Z}(\beta)=\int \mathrm{d}q \,\Pi(\hbar\beta/2,-\hbar\beta/2;q,q) =\int[\mathrm{d}q(\tau)]\,\mathrm{e}^{-\mathcal{S}(\mathbf{q})/\hbar} \qquad\text{sum over all }\; q(\hbar\beta/2)=q(-\hbar\beta/2) \;,\]
that is as a path integral integrating over all closed paths, that is, all paths satisfying the periodic boundary condition \(q(\hbar\beta/2)=q(-\hbar\beta/2).\)
Gaussian path integrals: The quantum harmonic oscillator
Gaussian path integrals, like finite dimensional Gaussian integrals, can be calculated explicitly. A more general example is provided by the path integral representation associated to the quantum harmonic oscillator defined by the quantum Hamiltonian \[\tag{26} \hat H=\frac{{\hat p}^2}{2 m}+\textstyle{\frac{1}{2} }m\;\omega^2\; {\hat{q} }^2\;,\]
where \(m\)is the mass of the particle and \(2\pi/\omega\) is the period of the classical oscillations. Then, the expression (12) becomes \[\mathcal {L}_{\mathrm{E}}(\dot{q},q;\tau)= \textstyle{\frac{1}{2}}m\left(\dot{q}^2+ \omega^2 q^2\right).\] Like in the example of the Brownian motion, to calculate the corresponding path integral one first solves the classical equation of motion (Euler-Lagrange equation corresponding to \(\mathcal {L}_{\mathrm{E}}\)) \[-\ddot q(\tau)+\omega^2q(\tau)=0\] with the path integral boundary conditions \(q(t/2)=q''\ ,\) \(q(-t/2)=q' .\) (Since the potential is time-independent, we can choose \(t'=-t/2\) and \(t''=t/2\) in (10).) The classical solution is \[q_c(\tau) ={1\over\sinh (\omega t)}\left[q'\sinh\bigl(\omega(t/2-\tau)\bigr)+ q'' \sinh\bigl(\omega(\tau+t/2)\bigr)\right]\] and the corresponding classical action is \[\mathcal{S}(\mathbf{q_c})={m\omega \over 2 \sinh \omega t}\left[ \left(q^{\prime 2}+q^{\prime\prime 2} \right) \cosh \omega t -2q' q'' \right] .\] One then changes variables \(q(\tau)\mapsto r(\tau)=q(\tau)-q_c(\tau)\ ,\) such that \(r(-t/2)=r(t/2)=0\ .\) Since \[\mathcal{S}(\mathbf{q})=\mathcal{S}(\mathbf{q_c})+\mathcal{S}(\mathbf{r}),\] the path integral becomes \[\langle q''|\boldsymbol{\Pi}(t/2,-t/2)|q'\rangle=\mathrm{e}^{- \mathcal{S}(\mathbf{q_c})/\hbar}\int[\mathrm{d}r(\tau)]\mathrm{e}^{-\mathcal{S}(\mathbf{r})/\hbar},\] The integral over \(r(\tau)\) yields a normalization constant. Its dependence on \(\omega\) can be determined by differentiating the path integral with respect to \(\omega\ .\) The final normalization can be determined by comparing with the Brownian motion \(\omega=0.\) One finds \[\int[\mathrm{d}r(\tau)]\mathrm{e}^{-\mathcal{S}(\mathbf{r})/\hbar}=\left({ m\omega \over 2\pi \hbar\, \sinh \omega t }\right)^{1/2}.\] The quantum partition function \(\mathcal{Z}(\beta)=\mathrm{tr} \,\mathrm{e}^{-\beta\hat H}\) (see (25)) follows: \[\tag{27} \mathcal{Z}(\beta)=\int\mathrm{d} q\,\langle q|\boldsymbol{\Pi}(\hbar\beta/2,-\hbar\beta/2)|q\rangle={1\over2\sinh(\beta\hbar\omega/2)} ={\mathrm{e}^{-\beta\hbar\omega/2} \over 1-\mathrm{e}^{-\beta\hbar\omega}}=\sum_{n=0}^{\infty}\mathrm{e}^{-\beta\hbar\omega(n+\frac{1}{2})}\;.\]
From (27) and the definition \(\mathcal{Z}(\beta)=\mathrm{tr} \,\mathrm{e}^{-\beta\hat H}\ ,\) one recovers the exact spectrum of the quantum Hamiltonian \( \hat H\) whose eigenvalues are \(E_n=\hbar\omega(n+\textstyle{\frac{1}{2}})\ .\)
Perturbative expansion for the path integral
Correlation functions
Generalizing to the interacting case the definition given for the free particle in the section "Discussion", correlation functions are the moments of the measure associated to the integrand in the path integral. They are defined by \[\tag{28} \langle q(\tau_1)q(\tau_2)\ldots q(\tau_{p})\rangle={1\over\mathcal{Z}}\int[\mathrm{d}q(\tau)]q(\tau_1)q(\tau_2)\ldots q(\tau_{p})\mathrm{e}^{-\mathcal{S}(\mathbf{q})/\hbar}\]
with \[\mathcal{Z}=\int[\mathrm{d}q(\tau)]\mathrm{e}^{-\mathcal{S}(\mathbf{q})/\hbar}.\] In the example of periodic boundary conditions \(q(-t/2)=q(t/2),\) (\(q(t/2)\) integrated), which correspond to a summation over all closed trajectories, \(\mathcal{Z}(\beta)\) is the quantum partition function describing the thermodynamical equilibrium at temperature \(1/\beta=\hbar/t\ .\)
Gaussian expectation values and Wick's theorem
We now consider path integrals corresponding to centred Gaussian measures, for which the action is a quadratic form in terms of the integration path \(q(\tau),\) (simple examples being provided by the Brownian motion and the quantum harmonic oscillator). Then, the symmetry \(q\mapsto -q\) implies that correlation functions (28) odd in \(q\) vanish. Moreover, for all centred Gaussian measures, correlation functions can be expressed in terms of the two-point function as stated by Wick's theorem (proved in a different context by Wick (1950)): \[\langle q(\tau_1)q(\tau_2)\ldots q(\tau_{2\ell}) \rangle=\sum _{ \scriptstyle{P \left\{ 1,2,\ldots 2\ell \right\}}}\langle q(\tau_{P_1})q(\tau_{P_2})\rangle\ldots \langle q(\tau_{P_{2\ell-1}}) q(\tau_{P_{2\ell}})\rangle,\] where \(P \left\{ 1,2,\ldots, 2\ell \right\}\) are all possible (unordered) pairings of \(\left\{ 1,2,\ldots, 2\ell \right\}\ .\) For example, \[\langle q(\tau_1)q(\tau_2)q(\tau_3)q(\tau_4)\rangle=\langle q(\tau_1)q(\tau_2)\rangle \langle q(\tau_3)q(\tau_4)\rangle +\langle q(\tau_1)q(\tau_3)\rangle \langle q(\tau_2)q(\tau_4)\rangle +\langle q(\tau_1)q(\tau_4)\rangle \langle q(\tau_3)q(\tau_2)\rangle.\] One proof of the theorem relies on introducing the generating functional of correlation functions \[\mathcal{Z}(j(\tau))=\langle \exp \int\!\mathrm{d}\tau \; j(\tau) q(\tau)\rangle,\] where \(j(\tau)\) is an arbitrary function. Correlations functions are obtained by taking multiple functional derivatives of \(\mathcal{Z}(j(\tau))\) with respect to \(j(\tau)\ .\) The generating functional \(\mathcal{Z}(j(\tau))\) can be calculated explicitly by Gaussian integration. Its expansion in powers of \(j(\tau)\) leads to Wick's theorem.
In the example of the quantum harmonic oscillator and periodic boundary conditions, in the limit \(t\to\infty\) the two-point function reduces to \(\langle q(\tau_1)q(\tau_2)\rangle \sim{\hbar\over2 m\omega} \mathrm{e}^{-\omega|\tau_1-\tau_2|}.\)
Perturbative expansion
We now assume that the potential \(V(q,t)\) in the path integral is a polynomial in \(q\ ,\) though perturbation theory can be generalized to analytic potentials. In the section Gaussian expectation values and Wick's theorem it has been shown that Gaussian expectation values can be calculated explicitly: therefore, to evaluate a path integral, a possible method is to keep the quadratic part (\(O(q^2)\)) of the potential of \(V(q,t)\) in the exponential as part of the Gaussian measure and to expand the remainder in a power series. For illustration purpose, we consider the quartic anharmonic oscillator, \[V(q)=\frac{1}{2} q^2+ \lambda q^4, \lambda >0\,.\] We set \[\mathcal{S}(\mathbf{q})=\mathcal{S}_0(\mathbf{q})+\lambda\int\mathrm{d}\tau \,q^4(\tau)\] with \[\mathcal{S}_0(\mathbf{q})=\frac{1}{2}\int\mathrm{d}\tau\,\bigl(\dot{q}^2(\tau)+q^2(\tau)\bigr).\] The expansion of the corresponding path integral can then be written as (here we set \(\hbar=1\)) \[\mathcal{Z}(\lambda)= \int[\mathrm{d}q(\tau)]\mathrm{e}^{-\mathcal{S}_0(\mathbf{q})}\sum_{k=0}^\infty {(-\lambda)^k\over k!}\left[\int\mathrm{d}\tau \,q^4(\tau)\right]^k\sim\mathcal{Z}(0)\sum_{k=0}^\infty {(-\lambda)^k\over k!}\left\langle \left[\int\mathrm{d}\tau \,q^4(\tau)\right]^k\right\rangle_0 ,\] where \(\langle\bullet\rangle_0\) means expectation value with respect to the Gaussian measure associated with \(\mathcal{S}_0\) and the symbol \(\sim\) means that the perturbative series is not convergent. Each term in the series can then be evaluated using Wick's theorem and the explicit form of the Gaussian two-point function. For example, at first order in \(\lambda\ ,\) \[\langle q^4(\tau)\rangle_0=3 (\langle q^2(\tau)\rangle_0)^2.\] The next order involves \[ \langle q^4(\tau)q^4(\tau')\rangle_0=9 (\langle q^2(\tau)\rangle_0)^2(\langle q^2(\tau')\rangle_0)^2+72 \langle q^2(\tau)\rangle_0 \langle q^2(\tau')\rangle_0(\langle q(\tau)q(\tau')\rangle_0)^2+24 (\langle q(\tau)q(\tau')\rangle_0)^4.\] It is then convenient to represent individual contributions graphically in terms of Feynman diagrams. Let us point out that such an expansion is divergent for all values of the parameter \(\lambda\ .\) It is an asymptotic series, useful as such only for \(\lambda\) small enough. For larger values of the expansion parameter, series summation methods are required.
Quantum time evolution
Following Feynman (Feynman 1948), quantum time-evolution (here we refer to real physical time) can be described in terms of (oscillatory) path integrals. In this formalism, considering a system classically described by the Cartesian coordinates \(\mathbf{q}\equiv\{q^1,q^2\ldots\}\ ,\) the matrix elements of the quantum evolution operator \(\mathbf{U}(t'',t') \) between times \( t'\) and \( t''\) are given by a sum over all possible trajectories (paths) \(\mathbf{q}(\tau)\equiv\{q^1(\tau),q^2(\tau)\ldots\}\ ,\) which in the simplest cases can be written as \[\tag{29} \langle \mathbf{q}'' \left|\mathbf{U}(t'',t') \right| \mathbf{q}' \rangle = \int \left [ \mathrm{d} \mathbf{q} (\tau) \right] \exp\left({i \over \hbar}\mathcal{A} (\mathbf{q} )\right) \]
with the boundary conditions \[\tag{30} \mathbf{q}(t')=\mathbf{q}' , \ \mathbf{q}(t'')=\mathbf{q}'',\]
where the classical action \( \mathcal{A} (\mathbf{q} ) \) is the time-integral of the classical Lagrangian: \[\tag{31} \mathcal{A}( \mathbf{q})=\int_{t'}^{t''}\mathrm{d} \tau\, \mathcal{L}\left( \mathbf{q}(\tau),\dot{\mathbf{q}}(\tau);\tau\right).\]
The expression (29) is valid when the kinetic term, that is, the term with two time-derivatives in the Lagrangian has the form \( \textstyle{\frac{1}{2}\sum_i} m_i(\dot{q}^i)^2 \ ,\) otherwise the measure has to be modified and new problems arise. An example of the latter situation is provided when the coordinates \(q^i\) parametrize a Riemannian manifold and the kinetic term involves \( \textstyle{\sum_{i,j}}\dot{q}^i g_{ij}(\mathbf{q})\dot{q}^j\ ,\) where \(g_{ij}\) is the metric tensor.
The formulation of quantum mechanics in terms of path integrals actually explains why equations of motion in classical mechanics can be derived from a variational principle. In the classical limit, that is, when the typical classical action is large with respect to \( \hbar\ ,\) the path integral can be evaluated by using the stationary phase method. The sum over paths is thus dominated by paths that leave the action stationary: the classical paths that satisfy \[\tag{32} \mathcal{A}\left(\mathbf{q}+\delta\mathbf{q}\right)-\mathcal{A}\left(\mathbf{q}\right)=O(\|\delta\mathbf{q}\|^2)\ \Rightarrow\ \frac{\delta \mathcal{A}}{\delta q^i}=0\ \Rightarrow\ {\partial\mathcal{ L}\over\partial q^i}-{\mathrm{d} \over\mathrm{d} t}{\partial\mathcal{L} \over\partial \dot{q}^i}=0 \]
with the boundary conditions (30). The leading order contribution is then obtained by expanding the path around the classical path, keeping only the quadratic term in the deviation and performing the corresponding Gaussian integration.
This property generalizes to relativistic quantum field theory.
From the mathematical point of view, it is much more difficult to define rigorously the real-time path integral than the imaginary-time statistical path integral. A possible strategy involves, when applicable, to calculate physical observables for imaginary time and then to proceed by analytic continuation.
Barrier penetration in the semi-classical limit
The purpose of this section is to illustrate with a simple example the evaluation of statistical (or imaginary time) path integrals in the semi-classical approximation. It is more technical and can be omitted in a first reading.
The path integral associated with \(\mathrm{tr}\,\mathrm{e}^{-t\hat H/\hbar}\ ,\) \[\tag{33} \mathcal{Z}=\int[\mathrm{d}q(\tau)]\mathrm{e}^{-\mathcal{S}(\mathbf{q})/\hbar} \quad\text{with}\; q(t/2)=q(-t/2),\]
is especially well suited to the evaluation, in the semi-classical limit \(\hbar\to 0\ ,\) of specific quantum phenomena called barrier penetration or tunnelling. Indeed, it can be shown that the classically forbidden barrier penetration appears, in the semi-classical limit, as formally related to classical evolution in imaginary time.
To explain the general idea, we consider an example of the form \[\tag{34} \mathcal{S}(\mathbf{q})=\int_{-t/2}^{t/2 }\mathrm{d}\tau\left[\frac{1}{2}m\dot{q}^2(\tau)+V\bigl(q(\tau)\bigr)\right], \]
with the potential \[\tag{35} V(q)=\frac{1}{2}q^2-\frac{1}{2}\lambda q^3\,,\]
which has one local minimum at \(q=0\) (\(V=0\)), a local maximum at \(q=2/(3\lambda)\) (\(V=4/(27\lambda^2)\)) and goes to \(\mp\infty\) when \(q\to \pm\infty\ .\)
The problem is to evaluate the probability per unit time for a particle localized initially in the well of the potential at \(q=0\) to escape the well. Since the potential (35) is not bounded from below, it is first necessary to define the quantum Hamiltonian. In this example, one can proceed by analytic continuation starting from \(\lambda\) pure imaginary. As conjectured initially by Bessis and Zinn-Justin (unpublished), the corresponding Hamiltonian, though complex, has a discrete real spectrum as a consequence of the symmetry \(q\mapsto-q\ ,\) \(\hat H\mapsto \hat H^*\ ,\) a symmetry also called PT symmetry (P being the parity transformation and T the time-reversal transformation). Returning by analytic continuation to \(\lambda\) real, one finds in this case a complex energy spectrum (quantum resonances), the imaginary part of the energy eigenvalues being directly related to tunnelling.
In the path integral framework, it can be shown that barrier penetration effects can be derived from an evaluation of the integral (33) for \(\hbar\to0\) and, therefore, by the steepest-descent method, suitably generalized to path integrals. One looks for non-trivial saddle points, here non-constant solutions of the classical equations of motion derived from the Euclidean action (34), which correspond formally to evolution in imaginary time. Moreover, if one is interested only in states with energies of order \( \hbar\ ,\) then one has to take the limit \(|t|\to\infty\ .\) Therefore, one looks for solutions that have a finite action on the real line. These solutions are called instantons. Here, the equation of motion obtained by varying \(\mathcal{S}\) is \[-\ddot q(\tau)+q(\tau)-\frac{3}{2}\lambda q^2(\tau)=0\,.\] Due to time translation invariance, one finds a one-parameter family of instanton solutions: \[q_c(\tau)={1\over\lambda \cosh^2((\tau-\tau_0)/2)}\ \Rightarrow\ \mathcal{S}(\mathbf{q}_c)=\int_{-\infty}^{+\infty}\mathrm{d}\tau \,\mathcal{L}_{\mathrm{E}}(\mathbf{q}_c)={8\over 15\lambda^2}.\] Completing the calculation of the saddle point contribution is a non-trivial exercise because it requires factorizing the path integral measure into an integration over \(\tau_0\) (a collective coordinate related to the breaking of time-translation symmetry by the solution) before using a saddle point approximation for the other modes of the path. One infers that, up to power law corrections, the probability per unit time of leaving the well is of order \( \exp{-\mathcal{S}(\mathbf{q}_c)/\hbar}=\exp{-8/(15\lambda^2\hbar)}.\)
Path integrals: Generalizations
We have presented only the simplest form of path integrals, which for the point of view of quantum mechanics involve only a classical Lagrangian with the general form (11). For more general Lagrangians or Hamiltonians, one encounters new problems in the definition of path integrals.
The quantum particle in a static magnetic field
When the Lagrangian involves a term linear in the velocity, as in the example of a quantum particle in a magnetic field, \[\tag{36} \mathcal{L}( \mathbf{q},\dot{\mathbf{q}})=\textstyle{1\over2 } \, m \,\dot{\mathbf{q}}^2-e \, \mathbf{A}( \mathbf{q})\cdot \dot {\mathbf{q}}\, ,\]
where \(\mathbf{A}( \mathbf{q})\) is a given vector potential, a new problem related to quantization arises. The classical Lagrangian together with the correspondence principle (replacing position and velocity by the corresponding quantum operators) does not determine the quantum theory because operators \(\mathbf{A}( \hat{\mathbf{q}}) \) and \(\dot{\hat{\mathbf{q}}}\) no longer commute. Correspondingly, the naive continuum form of the path integral is not defined because the continuum limit depends explicitly on the time-discretized form of the path integral and leads to a one-parameter family of different theories. This reflects, for example, in the appearance of undefined terms \(\operatorname{sgn}(0)\) in calculations. The underlying quantum Hamiltonian is then uniquely determined by demanding either its hermiticity or equivalently its gauge invariance. To determine the path integral, one can either return to a time-discretized form consistent with the quantum Hamiltonian (which implies the midpoint rule in the argument of the vector potential), or add a term with higher order time derivatives in the action, for example, \[\mathcal{S}\mapsto \mathcal{S}+\eta\int_{t'}^{t''}\mathrm{d}\tau(\ddot q(\tau))^2, \ \eta>0\,.\] This has the effect of restricting the integration to paths that satisfy a Hölder condition of order 3/2 and are thus differentiable, in such a way that expectations values with \(\dot{\mathbf{q}}\) are defined. This regularization does not violate gauge invariance but violates hermiticity of the Hamiltonian. In the case of the magnetic field, in the \(\eta\to0\) limit, it fixes the ambiguities (\(\operatorname{sgn}(0)=0\)) in a way that is consistent with a gauge invariant Hermitian quantum Hamiltonian.
Hamiltonian formulation and phase space integration
For a general classical Hamiltonian, the quantum evolution operator can formally be expressed in terms of a path integral involving an integration over phase space variables, position \( \mathbf{q} \) and conjugate momentum \( \mathbf{p}\ :\) \[\tag{37} \langle \mathbf{q}'' \left| \mathbf{U}(t'',t') \right| \mathbf{q}' \rangle = \int \left [ \mathrm{d} \mathbf{p} (\tau) \mathrm{d} \mathbf{q} (\tau)\right] \exp\left({i \over \hbar}\mathcal{A} (\mathbf{p},\mathbf{q} )\right) \]
with the boundary conditions \(\tag{38} \mathbf{q}(t')=\mathbf{q}' , \ \mathbf{q}(t'')=\mathbf{q}'',\)
where the classical action \( \mathcal{A} (\mathbf{p},\mathbf{q} ) \) is now expressed in terms of the classical Hamiltonian \(H(\mathbf{p},\mathbf{q};t):\) \[\tag{39} \mathcal{A}(\mathbf{p},\mathbf{q})=\int_{t'}^{t''}\mathrm{d} \tau\,\left[\mathbf{p}(\tau)\cdot\dot{\mathbf{q}}(\tau)- H\!\left(\mathbf{p}(\tau), \mathbf{q}(\tau); \tau\right)\right].\]
When the Hamiltonian is quadratic in the conjugate momentum \(\mathbf{p},\) the integral over \(\mathbf{p}(\tau)\) is Gaussian and can be performed explicitly: first one shifts \(\mathbf{p}(\tau)\) in the exponential by the solution of the classical equation \(\dot{\mathbf{q}}(\tau)=\frac{\partial H\!\left(\mathbf{p}(\tau), \mathbf{q}(\tau); \tau\right) }{\partial\mathbf{p}(\tau) } \ .\) One thus recovers in the exponential the classical Lagrangian. One then integrates over \(\mathbf{p}(\tau)\) and this may modify the \(\mathbf{q}(\tau)\)-integration measure if the coefficient of the quadratic term in \(\mathbf{p}(\tau)\) is not a constant. In the general case, the interpretation of this path integral reflects the problems of quantizing classical Hamiltonians and the order of operators in products. The Hamiltonian path integral has mainly a heuristic value (except in the semi-classical limit).
Holomorphic formalism and bosons
Up to now, we have described the path integral formalism relevant for distinct quantum particles. But quantum particles are either bosons, obeying the Bose-Einstein statistics or fermions, governed by Fermi-Dirac statistics. To describe the quantum evolution of several identical (and thus indiscernible) quantum particles, the path integral formulation has to be generalized. In the case of bosons, it is based on the coherent states holomorphic formalism and the Hilbert space of analytic entire functions. For bosons occupying only a finite number of quantum states, the relevant path integral can formally be deduced from the phase space integral by a complex change of variables, up to boundary terms and boundary conditions. In the example of one quantum state, the change of variables is simply \[z=(p+iq)/i\sqrt{2}\,,\quad \bar z=i(p-iq)/\sqrt{2}\,.\] The holomorphic path integral then takes the form \[\tag{40} \langle \mathbf{z}'' \left|\mathbf{U} (t'',t') \right| \bar{\mathbf{z}}' \rangle = \int \left [ \mathrm{d}\bar {\mathbf{z}} (\tau) \mathrm{d} \mathbf{z} (\tau)\right]\mathrm{e}^{\bar{\mathbf{z}}(t')\cdot\mathbf{z}(t')} \exp\left({i \over \hbar}\mathcal{A} (\mathbf{z},\bar{\mathbf{z}} )\right) \]
with the boundary conditions \(\tag{41} \bar{\mathbf{z}}(t')=\bar{\mathbf{z}}' , \ \mathbf{z}(t'')=\mathbf{z}'',\)
where the classical action \( \mathcal{A} (\mathbf{z},\bar{\mathbf{z}} ) \) reads \[\tag{42} \mathcal{A}(\mathbf{z},\bar{\mathbf{z}})=\int_{t'}^{t''}\mathrm{d} \tau\,\left[-i\bar{\mathbf{z}}(\tau)\cdot\dot{\mathbf{z}}(\tau)- H\left(i(\mathbf{z}(\tau)-\bar {\mathbf{z}}(\tau))/\sqrt{2} ,(\mathbf{z}(\tau)+\bar{\mathbf{z}}(\tau))/\sqrt{2} ; \tau\right)\right].\]
More generally, to \(N\) quantum states are associated \(N\) pairs of complex variables \((z_i,\bar{z_i})\ .\)
Even in the Gaussian example, this path integral suffers from the same ambiguities as in the example of a particle in a magnetic field, and this leads also to the appearance of \(\operatorname{sgn}(0)\) in calculations.
Grassmann path integrals and fermions
The understanding of this section necessitates some prior knowledge of Grassmann or exterior algebras, including the definition and properties of Grassmann differentiation and integration.
The description of the statistical properties or of the quantum evolution of fermion systems requires the introduction of elements of an infinite dimensional Grassmann algebra and the integration over Grassmannian paths. For example, to describe a system with \(N\) available quantum states, one introduces the generators \(\theta_i(\tau),\) \(\bar\theta_i(\tau),\) \(i=1\ldots N\ ,\) of a Grassmann algebra. They satisfy the commutation relations \[\theta_i(\tau)\theta_j(\tau')+\theta_j(\tau') \theta_i(\tau)=0\,,\quad \theta_i(\tau)\bar\theta_j(\tau')+\bar\theta_j(\tau') \theta_i(\tau)=0\,,\quad\bar\theta_i(\tau)\bar\theta_j(\tau')+\bar\theta_j(\tau') \bar\theta_i(\tau)=0\,.\] Then, rules of Grassmannian differentiation and integration can be formulated. It follows that the elements of the density matrix at thermal equilibrium, or the imaginary-time path integral, take the form \[\left\langle \boldsymbol{\theta} '' | U (t'' ,t' ) | \bar {\boldsymbol{\theta}}' \right\rangle = \int^{ \boldsymbol{\theta} (t'' )= \boldsymbol{\theta} ''}_{\bar {\boldsymbol{\theta}} (t' )=\bar {\boldsymbol{\theta}}'} \left[ \mathrm{d} \boldsymbol{\theta} (\tau)\mathrm{d}\bar {\boldsymbol{\theta}} (\tau) \right] \mathrm{e}^{- \bar{\boldsymbol{\theta}} (t')\cdot\boldsymbol{\theta} (t')} \exp\left[-\mathcal{S} ( \boldsymbol{\theta} ,\bar {\boldsymbol{\theta}} )\right] \] with \[\mathcal{S} ( \boldsymbol{\theta} , \bar{\boldsymbol{\theta}} ) = \int^{t''}_{t'} \mathrm{d} \tau\, \left\{ \bar{\boldsymbol{\theta}} (\tau)\cdot \dot {\boldsymbol{\theta}} (\tau)+H\left[ \boldsymbol{\theta} (\tau), \bar{\boldsymbol{\theta}} (\tau) \right] \right\} , \] where \(H\left[ \boldsymbol{\theta} (\tau), \bar{\boldsymbol{\theta}} (\tau) \right]\) represents the Hamiltonian acting on Grassmann functions.
A generalization: The field integral
While the path integral is an interesting topic for its own sake, the most useful physics applications are provided by a generalization: the field integral, where the integration over paths is replaced by an integration over fields. For example, in a local field theory for a neutral scalar field \(\phi(x),\) \(x\in\mathbb{R}^d\ ,\) the partition function is given by \[\mathcal{Z}=\int[\mathrm{d}\phi(x)]\mathrm{e}^{-\mathcal{S}(\phi)/\hbar},\] where the Euclidean action \(\mathcal{S}(\phi)\) is a \(d\)-dimensional integral of a function of the field and its derivatives (\(\partial_\mu\equiv \partial/\partial x_\mu\)): \[\mathcal{S}(\phi)=\int\mathrm{d}^d x\,\mathcal{L}_{\mathrm{E}}(\partial_\mu\phi(x),\phi(x)).\] While the algebraic properties of the path integral generalize easily, the field integral leads to new problems requiring new concepts like regularization and renormalization. Indeed, a natural physical choice, for example, would be \[\tag{43} \mathcal{L}_{\mathrm{E}}(\partial_\mu\phi,\phi)=\frac{1}{2}\sum_\mu(\partial_\mu\phi)^2+ \frac{1}{2}r\phi^2+{g \over 4!}\phi^4,\]
where \(r\) and \(g \ge 0\) are two parameters characterizing the model. However, in dimensions \(d>1\) the derivative term no longer selects fields regular enough, as a discrete or lattice approximation reveals, and, as a consequence, field correlation functions are not defined at coinciding points. It is necessary to modify (in an unphysical way from the viewpoint of quantum physics) the action at short distance, a procedure called regularization. One possibility is to introduce quadratic terms in the field with derivatives of higher order \(2n>d\ ,\) which restrict the integration to fields satisfying a Hölder condition as in \( d=1\ .\) Another possibility is to consider a lattice approximation with a lattice spacing \(1/\Lambda\ .\) The existence of a continuum limit or large \(\Lambda\) limit, then requires, in addition, tuning the initial parameters of the model as a function of \(\Lambda\ ,\) a procedure called renormalization. Renormalization, and its consequence, the renormalization group, find a natural interpretation in the theory of continuous macroscopic phase transitions.
In a relativistic-covariant quantum field theory, the real-time evolution (in 3+1 space-time dimensions) is then given by \[\int[\mathrm{d}\phi(x)]\exp\left[{i\over\hbar}\mathcal{A}(\phi)\right],\] where \(\mathcal{A}\) now is the classical action, space-time integral of the classical Lagrangian density \[\mathcal{A}(\phi)=\int\mathrm{d}^4 x\,\mathcal{L}(\partial_\mu\phi(x),\phi(x)).\] In the example of the Euclidean Lagrangian (43), the real-time Lagrangian reads (\(t\equiv x_0\)) \[\mathcal{L}=\frac{1}{2}(\partial_t \phi)^2-\frac{1}{2}(\nabla\phi)^2 -\frac{1}{2}r\phi^2-{g \over 4!}\phi^4,\] where \(\nabla\equiv \{\partial/\partial x^1,\partial/\partial x^2,\partial/\partial x^3\}\ .\)
Beside scalar boson fields, in general other types of fields are also required like Grassmann fields with spin for fermion matter. Moreover, since in the Standard Model that describes fundamental interactions at the microscopic scale, interactions are generated by the principle of gauge invariance, gauge fields also appear (and unphysical spinless fermions after quantization).
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Internal references
- James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
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- Jean Zinn-Justin and Riccardo Guida (2008) Gauge invariance. Scholarpedia, 3(12):8287.
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- James Meiss (2007) Hamiltonian systems. Scholarpedia, 2(8):1943.
- Andrei D. Polyanin, William E. Schiesser, Alexei I. Zhurov (2008) Partial differential equation. Scholarpedia, 3(10):4605.
- Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
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- Coleman, S (1979). The uses of instantons in The Whys of Subnuclear Physics, Erice 1977, Zichichi A ed. Plenum, New York. reprinted in Coleman S (1985) Aspects of Symmetry, Cambridge University Press, ISBN 0-521-31827-0
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- Faddeev, L D (1976). Les Houches School 1975 in Methods in Field Theory, Balian R and Zinn-Justin J eds. Elsevier Science Publishing, Amsterdam. ISBN 978-0720404333
- Faddeev, L D and Slavnov, A A (1991). Gauge Fields. Introduction to quantum theory. (2nd edition). Addison-Wesley Publishing Company, T. ISBN 0201524724.
- Grosche, C and Steiner, F (1998). Handbook of Feynman path integrals. Springer, Berlin, Heidelberg. ISBN 978-3540571353
- Kleinert, H (1995). Path Integrals in Quantum Mechanics, Statistics and Polymer Physics. World Scientific, Singapore. ISBN 978-981-270-009-4
- Popov, V N (2002). Functional Integrals in Quantum Field Theory and Statistical Physics, translated from Russian. Springer, Heildelberg. ISBN 978-1-4020-0307-3
- Schulman, L S (2005). Techniques and applications of path integration. Dover Publications, US. ISBN 978-0486445281
- Vasilev, A N (1998). Functional Methods in Quantum Field Theory and Statistical Physics, translated from Russian. Gordon and Breach, Amsterdam. ISBN 90-5699-035-7
- Zinn-Justin, J (2004). Path Integrals in Quantum Mechanics. Oxford University Press, Oxford. ISBN 978-0198566748
- French version: Intégrale de chemin en mécanique quantique, EDP Sciences 2003 ISBN 978-2868836601, Russian translation Fislitmat (Moscow 2006).
- Zinn-Justin, J (2002). Quantum Field Theory and Critical Phenomena (4th edition). Oxford University Press, Oxford. ISBN 0198509235
See also
Coherent state (quantum mechanics), Density matrix, Gauge invariance, Gauge theories, Instanton, Path integral (mathematical physics), Quantum mechanics, Schrödinger equation,


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Here at Chronically Vintage, it's scarcely a secret that I'm a big fan of The Best Vintage Clothing. They've a thriving vintage online empire and have been a longtime blog sponsor and giveaway partner. Today I'm pleased as punch to bring you another opportunity for your chance to win a $50.00 gift certificate for The Best Vintage Clothing's stand alone e-shop.
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subset: commoncrawl


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Ek Tha Tiger
In the dark world of intelligence and espionage... all information is guarded in the name of national security... But some stories escape the fiercely guarded classified files... stories that become legends. This is one such story... a story that shook the very foundation of this dark world. In a government employees' neighborhood in New Delhi, there lived a rugged, handsome and mysterious bachelor about whom his neighbors knew nothing. That was because he was India's top spy, an officer with the Research and Analysis Wing (RAW), India's external intelligence agency. This man was known even in official corridors as TIGER (Salman Khan). Tiger is sent on a relatively easy and safe mission to Dublin, Ireland to observe a scientist of Indian origin who is suspected of sharing his research findings with the Pakistan defense establishment. Tiger attempts to meet the scientist but is not successful. He tries to befriend the scientist's part time home caretaker, Zoya (Katrina Kaif). As Tiger begins to get closer to Zoya... he begins to discover his human side. For the first time in his life Tiger falls in love... What follows is a tumultuous journey that Tiger and Zoya embark upon, battling the dark world of intelligence and espionage that forbids its soldiers from falling in love.
Salman Khan, Katrina Kaif, Ranvir Shorey
Kabir Khan


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- This event has passed.
First Friday Art Walk
March 1, 2019 @ 6:00 pm - 9:00 pm
An event every month that begins at 6:00 pm on day First of the month, repeating indefinitely
Be inspired! ~ Art! ~ Shopping! ~ Entertainment!
The First Friday Art Walk takes place the first Friday of each month from 6 to 9 p.m. First Friday Art Walk is a free event located within Ocala's historic downtown. Event festivities include over 30 artist displays, monthly live entertainment, free family art activities provided by local arts organizations and extended shopping hours. The public is invited to enjoy an evening of art, live performances, and downtown shops and restaurants. Pick up your map on the square and take a self-guided tour of local artists displaying their works for sale in front of our downtown business. Stop to take in the performing artists stationed along the way.


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Annual The area of lowest rainfall is in the vicinity of Lake Eyre in South Australia, where the median annual rainfall is only about 100 mm. Another very low rainfall area is in Western Australia in the region of the Giles-Warburton Range, which has a median annual rainfall of about 150 mm. A vast region, extending from the west coast near Shark Bay across the interior of Western Australia and South Australia to south-west Queensland and north-west New South Wales, has a median annual rainfall of less than 200 mm. This region is not normally exposed to moist air masses for extended periods and rainfall is irregular, averaging only one or two days per month. However, in favourable synoptic situations, which occur infrequently over extensive parts of the region, up to 400 mm of rain may fall within a few days and cause widespread flooding. The region with the highest median annual rainfall is the east coast of Queensland between Cairns and Cardwell, where Happy Valley has a median of 4,436 mm (43 years from 1956 to 2000 inclusive) and Babinda a median of 4,092 mm (84 years from 1911 to 2000 inclusive). The mountainous region of western Tasmania also has a high annual rainfall, with Lake Margaret having a median of 3,565 mm (76 years to 1987 inclusive). The Snowy Mountains area in New South Wales also has a particularly high rainfall. While there are no gauges in the wettest area, on the western slopes above 1,800 metres elevation, runoff data suggest that the median annual rainfall in parts of this region exceeds 3,000 mm. Small pockets with median annual rainfall exceeding 2,500 mm also exist in the mountainous areas of north-east Victoria and some parts of the east coastal slopes. Map 1.5 shows average annual rainfall over the Australian continent. 1.5 AVERAGE ANNUAL RAINFALL Seasonal As outlined earlier, the rainfall pattern of Australia is strongly seasonal in character, with a winter rainfall regime in the south and a summer regime in the north. The dominance of rainfall over other climatic elements in determining the growth of specific plants in Australia has led to the development of a climatic classification based on two main parameters, median annual rainfall and the incidence of seasonal rainfall. Evaporation and the concept of rainfall effectiveness are taken into account to some extent in this classification, by assigning higher median annual rainfall limits to the summer zones than to the corresponding uniform and winter zones. The main features of the seasonal rainfall are:
Figure 1.6 comprises individual graphs showing the monthly rainfall for all capital cities, as well as for Alice Springs and Davis Base in Antarctica. Darwin shows the rainfall distribution pattern typical of the wet summer and dry winter seen in far northern Australia, and Brisbane the wet summer/relatively dry winter typical of southeastern Queensland. By contrast, Adelaide and Perth show the wet winter/dry summer pattern whereas Sydney, Melbourne, Canberra and Hobart show a relatively uniform pattern of rainfall throughout the year. Alice Springs shows a low rainfall pattern throughout the year typical of arid inland areas. Precipitation at Davis Base is mainly as snow, but is measured as water after melting. The pattern reflects the very low precipitation levels on the Antarctic continent. A rainday occurs when more than 0.2 mm of rain falls in 24 hours, usually from 9 a.m. to 9 a.m. the next day. The frequency of raindays exceeds 150 per year in much of Tasmania (with a maximum of over 250 in western Tasmania), southern Victoria, parts of the north Queensland coast and in the extreme south-west of Western Australia. Over most of the continent the frequency is less than 50 raindays per year. The area of low rainfall with high variability, extending from the north-west coast of Western Australia through the interior of the continent, has less than 25 raindays per year. In the high rainfall areas of northern Australia, the number of raindays is about 80 per year, but heavier falls occur in this region than in southern regions. The values in table 1.7 represent intensities over only small areas around the recording points because turbulence and exposure characteristics of the measuring gauge may vary over a distance of a few metres. The highest 24 hour (9 a.m. to 9 a.m.) falls are listed in table 1.8. Most of the very high 24 hour falls (above 700 mm) have occurred in the coastal strip of Queensland, where a tropical cyclone moving close to mountainous terrain provides ideal conditions for spectacular falls.
The highest annual rainfalls are listed by State/Territory in table 1.9.
Thunderstorms and hail A thunderday at a given location is a calendar day on which thunder is heard at least once. The average annual number of thunderdays varies from 88 per year near Darwin to less than 10 per year over parts of the southern regions. Convectional processes during the summer wet season cause high thunderstorm incidence in northern Australia. The generally high incidence of thunderdays (40-60 annually) over the eastern upland areas is caused mainly by orographic uplift of moist air streams. Hail, mostly of small size (less than 10 mm diameter), occurs with winter-spring cold frontal activity in southern Australia. Summer thunderstorms, particularly over the uplands of eastern Australia, sometimes produce large hail (greater than 10 mm diameter). Large hail capable of piercing light-gauge galvanised iron occurs at irregular intervals and sometimes causes widespread damage. Snow


text:
Dark Web
Energistically visualize premier solutions for standards compliant potentialities. Efficiently streamline vertical solutions via timely communities. Phosfluorescently plagiarize effective infomediaries via web-enabled markets. Rapidiously utilize long-term high-impact meta-services whereas exceptional convergence. Uniquely formulate parallel interfaces without value-added ROI.
Rapidiously deploy market.
Type: Icon Design
Client: Creattica
Link: by yukozlovskiy
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Lo siento, debes estar conectado para publicar un comentario.


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wogma rating: Add to that never-watched 'To Watch' list (?)
A bunch of non-actors pull off heists in the remake of The Italian Job. And there are new twists of course! It's an Abbas-Mustan film after all! And yet it is not as bad as the promos look.
Read moreAbhishek Bachchan, Bipasha Basu, Sonam Kapoor, Neil Nitin Mukesh, Bobby Deol, Sikander Kher - read these names in one breath and you should be looking up for jokes on twitter. I bet you there are some very good ones doing the rounds. Most of them are "stars" but none of them is twinkling too well, right now. So the audience is bound to be skeptical and thus have a laugh or two at the combination's expense.
Add to this mix, the names Abbas-Mastan and you guaranteed to get lots of style and almost equal guarantee of no substance. Pick any promotional video from the lot that is out so far, not one shot is non-stylized. For example, in this supposedly romantic song here, I don't see two people in love or falling in love, I see two gorgeous-looking people trying to oozing style out of every single pore.
I understand Players is a remake of The Italian Job. It might not have been the best film ever, but it was an enjoyable heist thriller. And instinctively, I couldn't help but squirm at the mention of Abbas-Mastan remaking it. Oh well, you cannot complain that they aren't breaking barriers. Just look at them pull down the gender bias at minute 1:10 in the trailer below. A woman perfectly parking a race car at high speed. There's your woman getting a substantial role!
The music has received lukewarm response, or rather it is nothing out of the ordinary for the music director. Pritam. I think we have witnessed the lamest theme song ever for a thriller. But don't miss out on lyrics like, 'This narcissism is a system to intimidate.' Now, isn't that a killer line?
Music Reviews:
Milliblog - "Pritam does what he's good at, admirably well, in Players."
Music Aloud - "6.5/10"
Bollyspice - "2.5/5"
Happy Sing - "Players is once again a lovely album from Pritam where he mostly creates what he is an expert at, with one or two new things here and there"
Dunkdaft - " So, a soundtrack with 2-3 nice tracks is not bad,after all, its for an action movie. "
rediff - "2.5/5"
- meeta, a part of the audience
Is Players, The Italian Job? It is. But, it is not. Then it is. And then again, it is not. Because, after all, it is an Abbas-Mustan product. The twists and turns turning into more turns and twists is a given. The lawn tennis that the characters play with their loyalties is also expected. You wonder whether the makers are screening a joke on themselves as the good thief and bad thief mock each other for being predictable. So what if we might not be able to make out which one of the four characters is the mole? We know one of them ought to be, and that is predictable enough, dear directors.
So, there is no sitting on the edge of your seat business. There is no thrilling chase that has you wrapped around itself. But, I must admit, that some of the twists made the plot a little more interesting, even if it was only for a few minutes. Players is also a lot less dramatized than I anticipated. And it is, thankfully, less stylized than the trailers made it out to be. But, that isn't saying too much, is it? I expected it to ooze substance-free form in every frame. So, it's not like you are going to miss the slo-mos nor is there a dearth of pauses after every third word for dramatic effect.
And since that's pretty much intentional, what chance do our actors have to give a decent performance? Not that we have too much hope left from this set of actors. A quick reminder - Abhishek Bachchan, Bipasha Basu, Sonam Kapoor, Neil Nitin Mukesh, Bobby Deol, Sikander Kher. It is a shame that I have not a good word to say about even Bachchan Jr. The rest are aging only worse with every film. And boy, did they over-estimate Neil Nitin Mukesh's potential to deliver a non-standard role.
The only reasonable actor in this ensemble is Omi Viadya who is comic-relief incarnate. A couple of one-liners have brilliant timing, and that's all it takes to remember him. The whole theater laughed out loud at a joke which has been done to death on sms/facebook/what-have-you! That's how remarkable Omi is.
But, like any other Abbas-Mustan film, you are guaranteed some exotic locales that are shot beautifully. A slick movie, Players surely is. In fact, they've chosen not so often visited Russia and New Zealand, and have gone as close to the North Pole as a film crew can go too. The sound also contributes in making you feel like you are a part of the action. There were a number of scenes where you can feel your seat pulsing with the beat. Yet, Hindi action thrillers are miles away from their Hollywood counterparts, despite having smart action and camera work. They lack originality. They remain just that - remakes of other films.
The problem also is the directors' strong belief in adding masala to suit the Indian palate. I don't get the insistence on adding song and dance, lackluster at that! Nor do I understand the rationale behind going out of the way to include comedy amidst tense drama. Just because it makes me laugh, it doesn't stop it from becoming a distraction from the flow of the story or a particular scene. And it only adds fidgety minutes to the run-time. One of the things that I like about The Italian Job is that it makes a team out of strangers because they have a common enemy. And fortunately, Players doesn't lose that flavor. Alongside there is a tidbit of romantic undercurrent. It's a good thing that it is kept in its place and there isn't much ado about it. Abbas-Mustan, of course, also add the extra load of twists. Only this time around there is a relatively better attempt at tying loose ends.
Players might work a slight shade better only because the expectations from it are so low. Watchable not must watch, like many other films running on TV.
- meeta, a part of the audience
Thumbs up, by Anna Vetticad, annavetticadgoes2themovies : ...I wish it hadn't lazily glossed over the intricacies of the plots hatched by Charlie Mascarenhas (Abhishek Bachchan) and his troopers. I wish the directors had dumped Pritam's tuneless songs. I wish some of the acting and dialogues had been better. But the truth is that despite all this, the pace of Players is such that I enjoyed it. ... full review
Thumbs up, by Nikhat Kazmi, Times of India : ...Your Bollywood ventures end up like cold cuts - stylishly done - are so different from piping hot goulash. ... full review
So-So, by Stefan S, A Nutshell Review : ...Those involving Bipasha Basu worked since the actress is known for being sultry although one can tell of her losing some weight and curves, while Sonam Kapoor seemed to have bulked up a little (for the better rather than being painfully pencil thin) but for her dances to be a little bit awkward when she was supposed to be sensual.... full review
So-So, by Rony D'Costa, Desi Martini : ...Should you watch it? - Yes (in the theatre with a bunch of friends) Why? Like I mentioned above Players is a bad film that entertains. ... full review
So-So, by Nupur Barua, fullhyd.com : ...The anorexic Sonam Kapoor in revealing clothes and Gothic make-up cannot dance, period. The less said about her insipid role, the better. She is too "Aisha" to be an intellectual hacker. ... full review
So-So, by Martin D'Souza, Glamsham.com : ...PLAYERS is a slick treat of high adrenaline action scenes that rates up there with the best in Bollywood.... full review
So-So, by Divya Solgama, Gomolo.in : ...Abhishek Bachchan has done plenty of films with conning / robbery as its main theme and out here he manages to do a decent job, though the charm of his last film 'Dum Maro Dum' is missing.... full review
So-So, by Shubha Shetty-Saha, MiD DAY : ...Bipasha looks sizzling hot, while Sonam fails to impress.... full review
So-So, by Stutee Ghosh, Planet Bollywood : ...While the actors, dialogue writers, editors and director duo draw flack for their horrid job, the cinematography and the manner in which the Russian landscape has been depicted and a couple of action sequences (although few and far between) hold us to the movie.... full review
So-So, by Shilpa Jamkhandikar, Reuters : ...It's all one-note, except for Neil Nitin Mukesh who is marginally better than the others. ... full review
So-So, by Ruchi Mehta, Roochster : ...Painfully long but entertaining... full review
So-So, by K K Rai, STARDUST : ...Bobby Deol, who plays the part of a magician and helps the players with his illusionist techniques in the heist, has largely been sidelined in the flick. He hardly gets any scope to showcase his histrionic ability.... full review
So-So, by SpiceZee Bureau, Zee News : ...Omi Vaidya is frustrating sometimes with his '3 Idiots' accent... full review
Thumbs down, by Naresh Kumar Deoshi, Apun Ka Choice : ...a bunch of non-actors, the esteemed coterie of Abhishek Bachchan, Bipasha Basu, Sonam Kapoor, Bobby Deol and Sikander Kher competing among themselves to see who hams less and fluffs least... full review
Thumbs down, by Bobby Sing, Bobby Talks Cinema.com : ...Bobby Deol, Omi Vaidya & Sikander Kher do not contribute anything significant in the film and are simply wasted... full review
Thumbs down, by Shalu Dhyani, Bolly Spice : ...Players is a good opportunity gone waste.... full review
Thumbs down, by Taran Adarsh, Bollywood Hungama : ...Pritam's music is, to put it bluntly, a minus point. ... full review
Thumbs down, by Charu A Mahesh, Buzz in Town : ...everything about the movie looked good and shouted attitude and oomph.... full review
Thumbs down, by Kshitij, Desi Martini : ...the only 2 stars who shine in this film are old warhorse Johnny Lever & Omi Vaidya. ... full review
Thumbs down, by Ameetbhuvan, Desi Martini : ...Players has some of the worst dialogues written in recent cinematic history, coupled with the most wooden cast ever.... full review
Thumbs down, by Yusuf Poonawala, Desi Martini : ...Besides a poor script and dialogue by Rohit Jugraj and Sudip Sharma, their screenplay is the deal breaker. Everything happens for the sake of convenience and too easily. ... full review
Thumbs down, by Ankur Pathak, Desi Martini : ...Players another heist-soup from the now-foreseeable style of director-brothers Abbas-Mustan has such an incompetent narrative, the actors in the motion picture react with an unintentionally true expression: Being Clueless. ... full review
Thumbs down, by Smita, Desi Martini : ...I have not seen that one but can bet my last plunging Rupee that it does not have thieves who give an excuse of setting up an orphanage to rob huge amounts of gold. It is this obstinacy of so many mainstream film makers to stick so closely to the formula that it makes a travesty of even the best plot. A crime thriller based on a moralistic premise is doomed from the start.... full review
Thumbs down, by Aniruddha Guha, DNA : ...This is B-movie paradise... full review
Thumbs down, by Deepa Gahlot, Film Impressions : ...All talk, lots of action, no fun.... full review
Thumbs down, by Deepa Deosthalee, Film Impressions : ...Omi Vaidya the miscast Punjabi make-up artist... full review
Thumbs down, by Mayank Shekhar, Hindustan Times : ......Which is good to know. But you still don't get the point. Well, hold on then, the banalities have merely begun; you'll stop questioning soon...... full review
Thumbs down, by Fatema H Kagalwala, India Entertainment : ...The only different thing that catches the attention, sometimes charmingly, mostly embarrassingly is Sonam Kapoor's experimental grunge look that the lady chose with little care but carries off with flair.... full review
Thumbs down, by Kaveree Bamzai, india today : ... What is Bobby Deol doing in the movie? No time for such existential questions. I was too distracted by Sonam Kapoor's wandering eye makeup and changeable hair.... full review
Thumbs down, by Shubhra Gupta, indian express : ...Even if you haven't seen the other two films, 'Players' comes off as a dismal first for Abbas-Mustan, with a flat script, flatter action scenes, and the flattest bunch of actors, all milling about in one film. ... full review
Thumbs down, by Komal Nahta, koimoi : ...Abhishek Bachchan and Bobby Deol, especially, seem to be under the mistaken belief that uttering dialogues in a very sober and serious way, with absolutely no smile or emotion on the face, is akin to good acting. But it only makes them look limited as actors.... full review
Thumbs down, by Sanjukta Sharma, Live Mint : ...Kapoor, Neil Nitin Mukesh, Deol and Kher are studies in stunted, contrived performances.... full review
Thumbs down, Movie Talkies : ...The bad plot points in the film are only accentuated by the even more terrible acting.... full review
Thumbs down, by Karan Anshuman, Mumbai Mirror : ...Good films are about bringing new ideas to viewers, and Players is antithetical in this respect. In the remake, the writers substitute what little logic was there to begin with, with songs and add deliberate cheesiness to make the film more accessible for the masses... full review
Thumbs down, Random Thoughts of a Demented Mind : ...Indeed. What a feast. Full of cheese.... full review
Thumbs down, by Sukanya Verma, Rediff : ...The keywords of this plot are simple: grand-scale gold robbery, fellowship of a highly-skilled team, successful execution, traitor in the gang, payback and a Mini Cooper-packed climax. ... full review
Thumbs down, by Sonia Chopra, Sify Movies : ...The Players race to the job... full review
Thumbs down, by Sudhish Kamath, The Hindu : ...Abeyaar, what is it to you? Ok, fine, we bought the rights, happy? But for the money we paid, we will remake both Italian Jobs in the same movie even if we just paid for one and set it everywhere but Italy. We will do unmentionable things to it right in front of your eyes and you can't complain we stole a film. ... full review
Thumbs down, by Reel Reptile, Upper Stall : ...- Bipasha Basu shows considerably more talent when she acts from the face down. ... full review
Thumbs down, by Kunal Guha, yahoo! India : ...Game Over... full review
Twitter reviews for this movie are not available.
This page has additional observations, other than the ones noted in the main review.
A bunch of strangers are put together to pull off a heist worth Rs.10,000 crore. All for an altruistic cause, of course. This then turns into a good-at-heart thief versus heartless thief battle. A battle of making fools out of each other.
Another rip off of a good movie. But the good part is they have honestly declared this as a remake and made it officially, unlike Sanjay Gupta's copies of Reservoir Dogs and Old Boy.
Meetu your review is very genuine and fair but i think the movie is not that bad compared to recent release as your verdict says.@Vinay yeah Sanjay Gupta's Kaante was indeed termed by tarantino as finest remake although being a unathorized remake .
Bollywood is now very addicted to remakes in name of inspired movie. I seriously don't like the this concept.Why don't we can have a original movie story . Every time i look for Hollywood movie for a good story .
@Vinay @Roy it's always going to be a tough call between badly made legal remake vs. well-made illegal copy.
@guddu i think that's what i just said, it is ok. i don't think it should be trashed all together.
@sadda adda :D


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The cold weather arrived early this fall in the Northeast and Midwest of the United States, which are the major natural gas consumers in this country. The arctic air marched through those regions, waking up the prices of the dormant natural gas, which have lagged oil prices for years. The price of one million British thermal units in Europe of this fuel is $12/MMBTU. This is three times the counter price in the United States. Meanwhile the price in Japan is $17/MMBTU, which is more than four times the price in the United States. But, the recent cold storm has increased the U.S. price to about $4.50/MMBTU, rising by 27 percent so far this year. It is the best performing commodity among the 22 commodities in the Dow Jones-UBS Commodity Index for the current fall, which is one of the coldest in more than a decade.
The cheapness and dormancy of natural gas prices has to do with the shale natural gas revolution that came after combining the vertical and horizontal drilling techniques to crack the shale formation. The revolution is a game changer that increased both the supply and reserves of this fuel in the United States and decreased its prices significantly at the household and firm levels.
It seems that the fall chill this year has awakened the natural gas price. It is expected to move up steadily but without spikes for many years to come. Its inventory has fallen this year to its lowest level since 2008. The projected colder temperatures for 2014 compared to the record-warm temperatures in 2012 should also bring higher natural gas prices than in that year. The steady upward movement in the price is also expected to continue in the next few years.
Additionally, the increase in demand (incremental demand) will continue to exceed the increase in supply in the next five years. This differential predicts steady increases but without jumps in natural gas prices. The wild card here is the speed of conversion of transportation fuels from crude oil to natural gas. Gas-to-liquids and compressed natural gas are predicted to absorb about three billion cubic feet per day of the increase in demand. There are companies that are now building the U.S. natural gas highway to be used particularly for transit buses and trucks.
In addition to the domestic demand driver coming from changes in domestic fundamentals, such as expected rising economic growth, a much stronger driver is expected to come from already-approved natural gas export terminals such as Cheniere Energy's. The export demand may reach 5.5 Bcf/d in the coming five years. By 2020, the United States could export as much as 61.7 million tons per year of liquefied natural gas. That would make the United States the second-largest LNG exporter in the world, next to Qatar. Other deals in the licensing program are likely to push the total closer to 80 million tons per year, which would exceed Qatar's current total of 77 million tons. This driver should be another push to natural gas prices over those years. China may play a similar role in driving up natural gas prices in the coming years as it has done with the crude oil prices over the last decade or so. China is building its power generation infrastructure and its power demand is expected to increase by ten times by 2035. Since natural gas is currently about 2 percent in the power fuel mix and there is a strong popular discontent with pollution, there should be a strong import demand for U.S. liquefied natural gas coming from China in the future.
The shale natural gas revolution is also real and a game changer for the U.S.'s energy costs. It has been felt in generating electricity at power plants and in warming American homes. It has also enhanced energy security as imports dropped and reserves increased significantly. Natural gas imports have been going down steadily. They should lead to improvements in the current account of the balance of payments.
In conclusion, people have been happy with the current $100 oil price. They should also be happy with natural gas prices ranging between $5-$6/MMBTU for the next few years.


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A business's first impression begins in the parking lot. That's why it's so crucial to put your parking lot paving in the hands of skilled and trained professionals. In Southern Philadelphia, your best bet for reliable parking lot services is Elite Paving & Concrete. We deliver top-notch parking lot construction results along with great value for your bottom line. Whether you're seeking parking lot pavers for a new site or parking lot contractors for expansions, repairs, or resurfacing, give us a call. We're the local parking lot company to count on for excellent service and flawless outcomes. Reach us today to discuss your parking lot paving needs in Upper Providence Township – Montgomery County PA!
Thanks to the parking lot pavers at Elite Paving & Concrete, you have one convenient source for a full range of options. As a result, you can save time and trouble by coming to us for all your parking lot services, no matter the size. Our commercial paving contractors have more than 50 years of firsthand experience. We're here to help with:
From initial grading to preparing the base levels to properly paving the entire surface, we'll see to each aspect of your parking lot construction. Our parking lot contractors even have you covered for the finishing touches, including stripes, curbs, speed bumps, and signs. Our goal is to make your parking lot installation and repairs go smoothly from step one to done. Not only can we adjust our parking lot services to suit your specific site, but we know how to bring you durable, weather-resistant asphalt paving that looks great and lasts long term.
Clear Parking Lot Striping Upper Providence Township – Montgomery County PA
Chaos can quickly ensue when parking lot lines are faint or missing altogether. Fortunately, Elite Paving & Concrete can carry out your parking lot striping in short order, clearly marking each space for safe and organized parking and traffic flow. Depending on your needs, we can stripe your parking lot as part of a new installation or as a separate service. Our parking lot striper team will customize your parking lot painting to reflect standard spaces, handicapped parking areas, reserved parking spots, and anything else you require. No parking lot construction is every really done until the lines are neatly painted on. We'll see your project through to the very end.
Effective Parking Stops Upper Providence Township – Montgomery County PA
Another way to help maintain an orderly parking lot is to install parking stops, otherwise known as parking curbs. Elite Paving & Concrete is ready to help with these features as well, giving drivers more to go on than parking lot lines alone. We can also help you keep out unwanted drivers by installing parking bumpers where cars are not permitted. Since each parking lot setup is different, it's a must to work with contractors who can accommodate your company's individual requests for parking stops, parking bumpers, and other key considerations. We're even able to tackle sign installations along with your parking blocks.
Traffic-Calming Speed Bumps Upper Providence Township – Montgomery County PA
Speed bumps have proven to be a surefire tool for getting a driver's attention and slowing down incoming vehicles. If you're interested in traffic-calming speed humps for your Fort Worth-area business, look no further than Elite Paving & Concrete. Whether all you need is a single speed bump or you'd like a succession of speed bumps for a long stretch of road, we're fully equipped to help. Speed hump installation is a simple yet effective way to force drivers to slow down. Don't overlook this essential safety factor when planning your parking lot. For more information about our speed bump installation service, reach us by phone or e-mail.


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Original Wood Floors, Inc is a family owned and operated company serving customers in the Miami-Dade County for more than 20 years. We have a great experience on hardwood floors installations, refinishing and repairs. Original Wood Floors, Inc offers hardwood floors installation, sanding, refinishing, and repairs to customers in Westwood Lakes, Florida. We certainly believe that everything must be about you "the customer".
Our main goal is always exceed our client's expectations and make sure everyone is satisfied at the end of the process. While always trying to make every transaction as simple as possible, our main priority is constantly improving our services from complicated wood floor installations and repairs to the simplest job.
You can not beat the beauty, warmth and elegance of a wooden floor. While hardwood is less durable than other materials, applying a urethane finish will help prevent the inevitable scratching that comes with this type of flooring. In addition, refinishing your hardwood (several times if necessary) is an easy way to bring your investment to life (something that is not possible with most other materials).
Laminate flooring is fast becoming one of the most sought after materials on the market. Laminate can mimic almost any surface, although different varieties of hardwood are by far the most popular. In addition, laminates are virtually impossible to scratch and highly resistant to staining and water.
The type of installation and materials your home needs is determined by your home being on a concrete slab or having a plywood subfloor. The two main ways of installing are referred to as a nailed down or a glued down installation. Each are effective and have their own unique qualities that make them particular to the type of subfloor you're installing on top of.
Hardwood floors add beauty and charm to any home. A hardwood floor that is scratched, dented, or excessively worn can take away from the attraction of a home and even decrease its value. Floors naturally become scuffed and fatigued over time. From dirt to high heels, UV sun exposure to your pet's untrimmed nails, your hardwoods durability is sure to wear over time.
When your hardwood floors get damaged, one of our experienced professionals can help you repair them the right way. Water damage, Fire damage, chipped boards, dented boards, cupping, cracking, or warping, we can fix it. Our team can make your hardwoods look new again. Matching an older floor to your new repair area takes skill and requires intensive labor.


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We offer one of the most comprehensive ranges of expansion joint covers in the world, backed by over 40 years design and manufacturing experience.
Our unparalleled technical support is available to help you at every stage from product selection right through to practical considerations during installation.
We want you to choose the right expansion joint cover and to help avoid costly mistakes our optional installation service is available for all of our products.


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Unified Storage in Virtual and Cloud Environments
Don't you wish there was a way to centralize management of block, file, and object data? Well, now there is.
Unified storage allows you to manage block, file and object data from a single foundation. This is becoming more and more advantageous as virtual server environments and the cloud force storage pros to look for consolidation and standardization wherever possible.
This expert E-Guide describes the architectural characteristics of unified storage and explains how it provides a measure of relief in virtual and cloud environments. Read now to explore the latest advances in this emerging technology space and what you can expect in terms of cost savings and management efficiency.