|
|
| Line 1: |
Line 1: |
| {{about|a formulation of quantum mechanics|integrals along a path, also known as line or contour integrals|line integral}}
| | Ugg Boots Extremely Shorter<br>To notify you the truth, the Ugg Boot has become the newest fashion merchandise that every single woman would seem to want. At present you will come across that the window displays of any outfits or shoe keep across the environment will have a pair of these trendy boots in it. It seems like Ugg Boots are right here to continue to be. <br>The simple fact that these boots are also currently being worn by the hottest Hollywood renowned men and women and supermodels is also a cash reward when it arrives to acquiring a span of them. Even now, if you desire to get additional than just a person pair of Ugg boots for fashion proposes and you would like to get them for marked-down Tolls so that you can preserve some revenue and get very good bargains, then there are affairs that you will have to figure out. The initially of these is that you won't be able to get your Ugg on sale if you never identify how to hold your eyes open for the <br>There are the pure colors (brown and sand) or you can be considerably a lot more daring, and say go for a pink or blue pair. In simple fact the decision appears to be to be infinite the place coloration is concerned. You could find you getting much more than one pair Ugg Boots that would match your female buddy of wife's mood flawlessly. If you needed to be recognized and shout listed here I am then Red or Pink could be the best shade for them, like Ugg Lo Pro Button beige (pink), but if you want men and women to assume they are sweet and girly, then might be you should invest in pink. On the other hand, if you just want them for comfort and ease and to insert a casual glimpse to their ensemble, then consider for a all-natural colour or even black. Whatsoever you decide you may well uncover that one particular pair is just not ample.<br>When picking which Ugg boots to invest in, it is vital to seem at a wide variety of suppliers, as costs can differ wildly. All people wishes to acquire a pair of decrease selling price and great top quality Ugg boots. You can even purchase a pair of these without leaving your very own property, as you will come across that there is numerous online web sites that are advertising this have to have product. As perfectly as the boots coming in normal colours, you can also get them in black, lilac, blue, pink and chestnut, which are just a handful of of the numerous distinctive shades accessible.<br>After all the decisions manufactured above you will need to look at what kind of cloth you want them manufactured out of, as now not only do the Ugg Boots come in a sheepskin model but they are also available in leather and suede. Having said that, will not forget about that only high high quality Ugg Boots are constantly lined with Australian Merino Sheepskin, and check out to keep away from the imitations that abound. Just one additional minor matter that any right minded woman should be pondering about when buying footwear not just Ugg Boots is the sole. With Ugg Boots you have the choice of large and clunky or clean soles. Obtain and get Ugg boots at uggfeet.com on the internet store, you can have a major favor! By the way, you can also MBT shoes below.<br><br>If you have any questions about where and how to use [http://tinyurl.com/k7shbtq http://tinyurl.com/k7shbtq], you can call us at our internet site. |
| {{Quantum mechanics|cTopic=Formulations}}
| |
| The '''path integral formulation''' of [[quantum mechanics]] is a description of quantum theory which generalizes the [[action (physics)|action principle]] of [[classical mechanics]]. It replaces the classical notion of a single, unique trajectory for a system with a sum, or [[functional integral]], over an infinity of possible trajectories to compute a [[probability amplitude|quantum amplitude]].
| |
| | |
| The basic idea of the path integral formulation can be traced back to [[Norbert Wiener]], who introduced the [[Wiener integral]] for solving problems in diffusion and [[Brownian motion]].<ref name=Demichev>
| |
| {{cite book |title=Path Integrals in Physics Volume 1: Stochastic Process & Quantum Mechanics |url=http://books.google.com/books?id=-XDP-8mrmQYC&pg=PA1 |chapter=Introduction |page=1 ''ff'' |isbn=0-7503-0801-X |year=2001 |publisher=Taylor & Francis |author=Masud Chaichian, Andrei Pavlovich Demichev}}
| |
| </ref> This idea was extended to the use of the [[Lagrangian]] in quantum mechanics by [[P. A. M. Dirac]] in his 1933 paper.<ref>{{cite journal |last=Dirac |first=Paul A. M. |authorlink=Paul Dirac |year=1933 |title=The Lagrangian in Quantum Mechanics |journal=Physikalische Zeitschrift der Sowjetunion |volume=3 |pages=64–72}}; also see
| |
| {{cite journal|title=The correspondence principle in the statistical interpretation of quantum mechanics | author=Van Vleck, John H | journal=Proceedings of the National Academy of Sciences of the United States of America|volume=14|number=2| pages=178188|year=1928|publisher=National Academy of Sciences}}</ref> The complete method was developed in 1948 by [[Richard Feynman]]. Some preliminaries were worked out earlier, in the course of his [[doctoral thesis]] work with [[John Archibald Wheeler]]. The original motivation stemmed from the desire to obtain a quantum-mechanical formulation for the [[Wheeler–Feynman absorber theory]] using a [[Lagrangian]] (rather than a [[Hamiltonian (quantum mechanics)|Hamiltonian]]) as a starting point.
| |
| | |
| This formulation has proven crucial to the subsequent development of [[theoretical physics]], because it is manifestly symmetric between time and space. Unlike previous methods, the path-integral allows a physicist to easily change [[coordinates]] between very different [[Canonical (disambiguation)#Physics|canonical]] descriptions of the same quantum system.
| |
| | |
| The path integral also relates quantum and [[stochastic]] processes, and this provided the basis for the grand synthesis of the 1970s which unified [[quantum field theory]] with the [[statistical mechanics|statistical field theory]] of a fluctuating field near a [[second-order phase transition]]. The [[Schrödinger equation]] is a [[diffusion equation]] with an imaginary diffusion constant, and the path integral is an [[analytic continuation]] of a method for summing up all possible [[random walk]]s. For this reason path integrals were used in the study of [[Brownian motion]] and [[diffusion]] a while before they were introduced in quantum mechanics.<ref>{{cite book |author=Kleinert, H. |year=1989 |title=Gauge Fields in Condensed Matter |volume=1: Superflow and Vortex Lines |chapter=6 |publisher=World Scientific |place=Singapore |isbn=9971-5-0210-0 |url=http://www.physik.fu-berlin.de/~kleinert/kleiner_reb1/contents1.html}}</ref>
| |
| | |
| [[Image:Three paths from A to B.png|thumbnail|250px|These are just three of the paths that contribute to the quantum amplitude for a particle moving from point A at some time ''t''<sub>0</sub> to point B at some other time ''t''<sub>1</sub>.]]
| |
| | |
| == Quantum action principle ==
| |
| | |
| In quantum mechanics, as in classical mechanics, the [[Hamiltonian (quantum mechanics)|Hamiltonian]] is the generator of time-translations. This means that the state at a slightly later time differs from the state at the current time by the result of acting with the Hamiltonian operator (multiplied by the negative [[imaginary unit]], −''i''). For states with a definite energy, this is a statement of the [[De Broglie relation]] between frequency and energy, and the general relation is consistent with that plus the [[superposition principle]].
| |
| | |
| But the Hamiltonian in classical mechanics is derived from a [[Lagrangian]], which is a more fundamental quantity considering [[special relativity]]. The Hamiltonian tells you how to march forward in time, but the time is different in different reference frames. So the Hamiltonian is different in different frames, and this type of symmetry is not apparent in the original formulation of quantum mechanics.
| |
| | |
| The Hamiltonian is a function of the position and momentum at one time, and it tells you the position and momentum a little later. The Lagrangian is a function of the position now and the position a little later (or, equivalently for infinitesimal time separations, it is a function of the position and velocity). The relation between the two is by a [[Legendre transform]], and the condition that determines the classical equations of motion (the [[Euler–Lagrange equation]]s) is that the action is a minimum.
| |
| | |
| In quantum mechanics, the Legendre transform is hard to interpret, because the motion is not over a definite trajectory. So what does the Legendre transform mean? In classical mechanics, with discretization in time,
| |
| | |
| :<math> \epsilon H = p(t)(q(t+\epsilon) - q(t)) - \epsilon L \,</math>
| |
| and
| |
| :<math> p = {\partial L \over \partial \dot{q} } \,</math>
| |
| | |
| where the partial derivative with respect to ''q'' holds ''q''(''t'' + ε) fixed. The inverse Legendre transform is:
| |
| | |
| :<math> \epsilon L = \epsilon p \dot{q} - \epsilon H \,</math>
| |
| | |
| where
| |
| | |
| :<math> \dot q = {\partial H \over \partial p} \,</math>
| |
| | |
| and the partial derivative now is with respect to ''p'' at fixed ''q''.
| |
| | |
| In quantum mechanics, the state is a [[quantum superposition|superposition of different states]] with different values of ''q'', or different values of ''p'', and the quantities ''p'' and ''q'' can be interpreted as noncommuting operators. The operator ''p'' is only definite on states that are indefinite with respect to ''q''. So consider two states separated in time and act with the operator corresponding to the Lagrangian:
| |
| | |
| :<math> e^{i( p (q(t+\epsilon) - q(t)) - \epsilon H(p,q) )}\,</math>
| |
| | |
| If the multiplications implicit in this formula are reinterpreted as matrix multiplications, what does this mean?
| |
| | |
| It can be given a meaning as follows: The first factor is
| |
| | |
| :<math> e^{-ip q(t)} \,</math>
| |
| | |
| If this is interpreted as doing a ''matrix'' multiplication, the sum over all states integrates over all ''q''(''t''), and so it takes the [[Fourier transform]] in ''q''(''t''), to change basis to ''p''(''t''). That is the action on the Hilbert space – '''change basis to p at time t'''.
| |
| | |
| Next comes:
| |
| :<math>e^{-i\epsilon H(p,q)} \,</math>
| |
| | |
| or '''evolve an infinitesimal time into the future'''.
| |
| | |
| Finally, the last factor in this interpretation is
| |
| | |
| :<math> e^{i p q(t+\epsilon)} \,</math>
| |
| | |
| which means '''change basis back to q at a later time'''.
| |
| | |
| This is not very different from just ordinary time evolution: the ''H'' factor contains all the dynamical information – it pushes the state forward in time. The first part and the last part are just doing Fourier transforms to change to a pure ''q'' basis from an intermediate ''p'' basis.
| |
| | |
| Another way of saying this is that since the Hamiltonian is naturally a function of ''p'' and ''q'', exponentiating this quantity and changing basis from ''p'' to ''q'' at each step allows the matrix element of ''H'' to be expressed as a simple function along each path. This function is the quantum analog of the classical action. This observation is due to [[Paul Dirac]].
| |
| {{quote box|align=right|width=44em|quote="...we see that the integrand in (11) must be of the form {{math|e<sup>i''F''/h</sup>}} where ''F'' is a function of {{math|''q''<sub>T</sub>,''q''<sub>1</sub>,''q''<sub>2</sub> ... ''q''<sub>m</sub>,''q''<sub>t</sub>}}, which remains finite as ''h'' tends to zero. Let us now picture one of the intermediate ''q''s, say ''q''<sub>k</sub>, as varying continuously while the other ones are fixed. Owing to the smallness of ''h'', we shall then in general have ''F/h'' varying extremely rapidly. This means that {{math|e<sup>i''F''/h</sup>}} will vary periodically with a very high frequency about the value zero, as a result of which its integral will be practically zero. The only important part in the domain of integration of ''q''<sub>k</sub> is thus that for which a comparatively large variation in ''q''<sub>k</sub> produces only a very small variation in ''F''. This part is the neighbourhood of a point for which ''F'' is stationary with respect to small variations in ''q''<sub>k</sub>. We can apply this argument to each of the variables of integration ....and obtain the result that the only important part in the domain of integration is that for which ''F'' is stationary for small variations in all intermediate ''q''s. ...We see that ''F'' has for its classical analogue {{math|{{intmath|int|T|t}} ''L dt''}}, which is just the action function which classical mechanics requires to be stationary for small variations in all the intermediate ''q''s. This shows the way in which equation (11) goes over into classical results when ''h'' becomes extremely small." |source=''Dirac (1932) op. cit., p. 69''}}
| |
| | |
| Dirac further noted that one could square the time-evolution operator in the S representation
| |
| | |
| :<math> e^{i\epsilon S} \,</math>
| |
| | |
| and this gives the time evolution operator between time ''t'' and time ''t'' + 2ε. While in the ''H'' representation the quantity that is being summed over the intermediate states is an obscure matrix element, in the S representation it is reinterpreted as a quantity associated to the path. In the limit that one takes a large power of this operator, one reconstructs the full quantum evolution between two states, the early one with a fixed value of ''q''(0) and the later one with a fixed value of ''q''(''t''). The result is a sum over paths with a phase which is the quantum action. Crucially, Dirac identified in this paper the deep quantum mechanical reason for the [[principle of least action]] controlling the classical limit (see quote box).
| |
| | |
| == Feynman's interpretation ==
| |
| | |
| Dirac's work did not provide a precise prescription to calculate the sum over paths, and he did not show that one could recover the Schrödinger equation or the [[canonical commutation relation]]s from this rule. This was done by Feynman.<ref>Both noted that in the limit of action that is large compared to the reduced [[Planck's constant]] ''ħ'' (using [[natural units]], ''ħ'' = 1), the path integral is dominated by solutions which are in the neighbourhood of [[stationary point]]s of the action.</ref>
| |
| | |
| Feynman showed that Dirac's quantum action was, for most cases of interest, simply equal to the classical action, appropriately discretized. This means that the classical action is the phase acquired by quantum evolution between two fixed endpoints. He proposed to recover all of quantum mechanics from the following postulates:
| |
| | |
| # The [[probability]] for an event is given by the squared length of a complex number called the "probability amplitude".
| |
| # The [[probability amplitude]] is given by adding together the contributions of all paths in configuration space.
| |
| # The contribution of a path is proportional to <math> e^{i S/\hbar}</math>. while ''S'' is the [[Action (physics)|action]] given by the [[time integral]] of the [[Lagrangian]] along the path.
| |
| | |
| In order to find the overall probability amplitude for a given process, then, one adds up, or [[integral|integrates]], the amplitude of postulate 3 over the space of ''all'' possible paths of the system in between the initial and final states, including those that are absurd by classical standards. In calculating the amplitude for a single particle to go from one place to another in a given time, it is correct to include paths in which the particle describes elaborate [[curlicues]], curves in which the particle shoots off into outer space and flies back again, and so forth. The '''path integral''' assigns to all these amplitudes ''equal weight'' but varying [[phase (waves)|phase]], or argument of the [[complex number]]. Contributions from paths wildly different from the classical trajectory may be suppressed by [[Interference (wave propagation)|interference]] (see below).
| |
| | |
| Feynman showed that this formulation of quantum mechanics is equivalent to the [[Quantization (physics)|canonical approach to quantum mechanics]] when the Hamiltonian is quadratic in the momentum. An amplitude computed according to Feynman's principles will also obey the [[Schrödinger equation]] for the [[Hamiltonian (quantum mechanics)|Hamiltonian]] corresponding to the given action.
| |
| | |
| == Concrete formulation ==
| |
| | |
| Feynman's postulates can be interpreted as follows:
| |
| | |
| === Time-slicing definition ===
| |
| For a particle in a smooth potential, the path integral is approximated by [[zig-zag]] paths, which in one dimension is a product of ordinary integrals. For the motion of the particle from position ''x<sub>a</sub>'' at time ''t<sub>a</sub>'' to ''x<sub>b</sub>'' at time ''t<sub>b</sub>'', the time sequence
| |
| :<math>t_a=t_0<t_1<...<t_{n-1}<t_n<t_{n+1}=t_b</math>
| |
| can be divided up into ''n'' + 1 little segments ''t<sub>j</sub>'' − ''t<sub>j − 1</sub>'', where ''j'' = 1,...,''n'' + 1, of fixed duration
| |
| :<math>\epsilon = \Delta t=\tfrac{t_b-t_a}{n+1}\,.</math>
| |
| | |
| This process is called ''time-slicing''.
| |
| | |
| An approximation for the path integral can be computed as proportional to
| |
| | |
| :<math>\int\limits_{-\infty}^{+\infty}\,\ldots \int\limits_{-\infty}^{+\infty}\,
| |
| \ \exp \left(\frac{{\rm i}}{\hbar}\int\limits_{t_a}^{t_b} L(x(t),v(t), t)\,\mathrm{d}t\right)dx_0 \, \ldots \, dx_n </math>
| |
| | |
| where <math> L(x,v,t)</math> is the Lagrangian of the 1d system with position variable ''x''(''t'') and velocity ''v'' = ''ẋ''(''t'') considered (see below), and ''dx<sub>j</sub>'' corresponds to the position at the ''j''th time step, if the time integral is approximated by a sum of ''n'' terms.<ref group="note">For a simplified, step by step, derivation of the above relation see [http://www.quantumfieldtheory.info/Path_Integrals_in_Quantum_Theories.htm Path Integrals in Quantum Theories: A Pedagogic 1st Step] [http://www.quantumfieldtheory.info/Path_Integrals_in_Quantum_Theories.pdf pdf vers]</ref>
| |
| | |
| In the limit ''n'' → ∞, this becomes a [[functional integral]], which – apart from a nonessential factor – is directly the product of the probability amplitudes <math>\langle x_a,t_a|x_b, t_b\rangle</math> – more precisely, since one must work with a continuous spectrum, the respective densities – to find the quantum mechanical particle at ''t<sub>a</sub>'' in the initial state ''x<sub>a</sub>'' and at ''t<sub>b</sub>'' in the final state ''x<sub>b</sub>''.
| |
| | |
| Actually <math>L</math> is the classical [[Lagrangian]] of the one-dimensional system considered, also
| |
| :<math> L(x,\dot x , t)=p\cdot \dot x - H(x,p,t)\,,</math>
| |
| where <math> H</math> is the Hamiltonian,
| |
| :<math> p=\frac {\partial L}{\partial \dot x} </math>, and the above-mentioned "zigzagging" corresponds to the appearance of the terms:
| |
| | |
| :<math>\exp\left (\frac{{\rm i}}{\hbar}\epsilon\, \,\sum_{j=1}^{n} L \left (\tilde x_{j},\frac{x_j-x_{j-1}}{\epsilon},j \right )\right )</math>
| |
| | |
| In the Riemannian sum approximating the time integral, which are finally integrated over ''x<sub>1</sub>'' to ''x<sub>n</sub>'' with the integration measure ''dx<sub>1</sub>...dx<sub>n</sub>x̃<sub>j</sub>'' is an arbitrary value of the interval corresponding to ''j'', e.g. its center, (''x<sub>j</sub> + ''x''<sub>''j'' − 1</sub>)/2.
| |
| | |
| Thus, in contrast to classical mechanics, not only does the stationary path contribute, but actually all virtual paths between the initial and the final point also contribute.
| |
| | |
| [[File:Path integral example.webm|thumb|The diagram shows the contribution to the path integral of a free particle for a set of paths.]]
| |
| | |
| Feynman's time-sliced approximation does not, however, exist for the most important quantum-mechanical path integrals of atoms, due to the singularity of the [[Coulomb potential]] ''e''<sup>2</sup>/''r'' at the origin. Only after replacing the time ''t'' by another path-dependent pseudo-time parameter
| |
| :<math>s=\int \frac{dt}{r(t)}</math>
| |
| the singularity is removed and a time-sliced approximation exists, that is exactly integrable, since it can be made harmonic by a simple coordinate transformation, as discovered in 1979 by [[İsmail Hakkı Duru]] and [[Hagen Kleinert]].<ref>{{cite journal |first=H |last=Duru |coauthors=Hagen Kleinert |title=Solution of the path integral for the H-atom |date=1979-06-18 |journal=Physics Letters |volume=84B |issue=2 |url=http://www.physik.fu-berlin.de/~kleinert/kleiner_re65/65.pdf |accessdate=2007-11-25}}</ref><ref>For details see Chapter 13 in [http://www.physik.fu-berlin.de/~kleinert/kleiner_reb5/psfiles/pthic13.pdf Kleinert's book] cited above.</ref> The combination of a path-dependent time transformation and a coordinate transformation is an important tool to solve many path integrals and is called generically the [[Duru–Kleinert transformation]].
| |
| | |
| === Free particle ===
| |
| | |
| The path integral representation gives the quantum amplitude to go from point x to point y as an integral over all paths. For a free particle action (''m'' = 1, ''ħ'' = 1):
| |
| :<math>
| |
| S= \int {\dot{x}^2\over 2} dt
| |
| \,</math>
| |
| | |
| the integral can be evaluated explicitly.
| |
| | |
| To do this, it is convenient to start without the factor ''i'' in the exponential, so that large deviations are suppressed by small numbers, not by cancelling oscillatory contributions.
| |
| :<math>
| |
| K(x-y;T) = \int_{x(0)=x}^{x(T)=y} \exp\left\{-\int_0^T {\dot{x}^2\over 2} dt\right\} Dx
| |
| \,</math>
| |
| | |
| Splitting the integral into time slices:
| |
| :<math>
| |
| K(x,y;T) = \int_{x(0)=x}^{x(T)=y} \Pi_t \exp\left\{-{1\over 2} \left({x(t+\epsilon) -x(t) \over \epsilon}\right)^2 \epsilon \right\} Dx
| |
| \,</math>
| |
| | |
| where the ''Dx'' is interpreted as a finite collection of integrations at each integer multiple of ε. Each factor in the product is a Gaussian as a function of ''x''(''t'' + ε) centered at ''x''(''t'') with variance ε. The multiple integrals are a repeated [[convolution]] of this Gaussian ''G''<sub>ε</sub> with copies of itself at adjacent times.
| |
| :<math>
| |
| K(x-y;T) = G_\epsilon*G_\epsilon ... *G_\epsilon
| |
| \,</math>
| |
| | |
| Where the number of convolutions is ''T''/ε. The result is easy to evaluate by taking the fourier transform of both sides, so that the convolutions become multiplications.
| |
| :<math>
| |
| \tilde{K}(p;T) = \tilde{G}_\epsilon(p)^{T/\epsilon}
| |
| \,</math>
| |
| | |
| The Fourier transform of the Gaussian ''G'' is another Gaussian of reciprocal variance:
| |
| | |
| :<math>
| |
| \tilde{G}_\epsilon(p) = e^{-\epsilon {p^2/2} }
| |
| \,</math>
| |
| | |
| and the result is:
| |
| :<math>
| |
| \tilde{K}(p;T) = e^{-T {p^2/2}}
| |
| \,</math>
| |
| | |
| The Fourier transform gives ''K'', and it is a Gaussian again with reciprocal variance:
| |
| :<math>
| |
| K(x-y;T) \propto e^{ -{(x-y)^2/(2T)}}
| |
| \,</math>
| |
| | |
| The proportionality constant is not really determined by the time slicing approach, only the ratio of values for different endpoint choices is determined. The proportionality constant should be chosen to ensure that between each two time-slices the time-evolution is quantum-mechanically unitary, but a more illuminating way to fix the normalization is to consider the path integral as a description of a stochastic process.
| |
| | |
| The result has a probability interpretation. The sum over all paths of the exponential factor can be seen as the sum over each path of the probability of selecting that path. The probability is the product over each segment of the probability of selecting that segment, so that each segment is probabilistically independently chosen. The fact that the answer is a Gaussian spreading linearly in time is the [[central limit theorem]], which can be interpreted as the first historical evaluation of a statistical path integral.
| |
| | |
| The probability interpretation gives a natural normalization choice. The path integral should be defined so that:
| |
| :<math>
| |
| \int K(x-y;T) dy = 1
| |
| \,</math>
| |
| | |
| This condition normalizes the Gaussian, and produces a Kernel which obeys the diffusion equation:
| |
| :<math>
| |
| {d\over dt} K(x;T) = {\nabla^2 \over 2} K
| |
| \,</math>
| |
| | |
| For oscillatory path integrals, ones with an ''i'' in the numerator, the time-slicing produces convolved Gaussians, just as before. Now, however, the convolution product is marginally singular since it requires careful limits to evaluate the oscillating integrals. To make the factors well defined, the easiest way is to add a small imaginary part to the time increment <math>\epsilon</math>. This is closely related to [[Wick rotation]]. Then the same convolution argument as before gives the propagation kernel:
| |
| :<math>
| |
| K(x-y;T) \propto e^{i(x-y)^2 / (2T)}
| |
| \,</math>
| |
| | |
| Which, with the same normalization as before (not the sum-squares normalization – this function has a divergent norm), obeys a free Schrödinger equation
| |
| :<math>
| |
| {d\over dt} K(x;T) = {\rm i} {\nabla^2 \over 2} K
| |
| \,</math>
| |
| | |
| This means that any superposition of ''K'''s will also obey the same equation, by linearity. Defining
| |
| :<math>
| |
| \psi_t(y) = \int \psi_0(x) K(x-y;t) dx = \int \psi_0(x) \int_{x(0)=x}^{x(t)=y} e^{iS} Dx
| |
| \,</math>
| |
| | |
| then ψ''<sub>t</sub>'' obeys the free Schrödinger equation just as ''K'' does:
| |
| :<math>
| |
| {\rm i}{\partial \over \partial t} \psi_t = - {\nabla^2\over 2} \psi_t
| |
| \,</math>
| |
| | |
| === The Schrödinger equation ===
| |
| | |
| {{Main|Relation between Schrödinger's equation and the path integral formulation of quantum mechanics}}
| |
| | |
| The path integral reproduces the Schrödinger equation for the initial and final state even when a potential is present. This is easiest to see by taking a path-integral over infinitesimally separated times.
| |
| :<math>
| |
| \psi(y;t+\epsilon) = \int_{-\infty}^\infty \;\;\psi(x;t)\int_{x(t)=x}^{x(t+\epsilon)=y} e^{{\rm i}\int_t^{t+\epsilon} (\frac{\dot{x}^2}{2} - V(x)) dt} Dx(t) \, dx
| |
| \qquad (1)</math>
| |
| | |
| Since the time separation is infinitesimal and the cancelling oscillations become severe for large values of ''ẋ'', the path integral has most weight for ''y'' close to ''x''. In this case, to lowest order the potential energy is constant, and only the kinetic energy contribution is nontrivial. The exponential of the action is
| |
| :<math>
| |
| e^{-{\rm i}\epsilon V(x)} e^{{\rm i}\frac{\dot{x}^2}{2}\epsilon}
| |
| \,</math>
| |
| | |
| The first term rotates the phase of ψ(''x'') locally by an amount proportional to the potential energy. The second term is the free particle propagator, corresponding to ''i'' times a diffusion process. To lowest order in ε they are additive; in any case one has with (1):
| |
| | |
| :<math>
| |
| \psi(y;t+\epsilon) \approx \int \psi(x;t) e^{-{\rm i}\epsilon V(x)} e^{{\rm i}(x-y)^2 \over 2\epsilon} dx
| |
| \,.</math>
| |
| | |
| As mentioned, the spread in ψ is diffusive from the free particle propagation, with an extra infinitesimal rotation in phase which slowly varies from point to point from the potential:
| |
| :<math>
| |
| \frac{\partial\psi}{\partial t} = {\rm i}\cdot \left[ \frac{1}{2}\nabla^2 - V(x)\right]\psi
| |
| \,</math>
| |
| | |
| and this is the Schrödinger equation. Note that the normalization of the path integral needs to be fixed in exactly the same way as in the free particle case. An arbitrary continuous potential does not affect the normalization, although singular potentials require careful treatment.
| |
| | |
| === Equations of motion ===
| |
| | |
| Since the states obey the Schrödinger equation, the path integral must reproduce the Heisenberg equations of motion for the averages of ''x'' and ''ẋ'' variables, but it is instructive to see this directly. The direct approach shows that the expectation values calculated from the path integral reproduce the usual ones of quantum mechanics.
| |
| | |
| Start by considering the path integral with some fixed initial state
| |
| :<math>
| |
| \int \psi_0(x) \int_{x(0)=x} e^{{\rm i}S(x,\dot{x})} Dx
| |
| \,</math>
| |
| | |
| Now note that ''x''(''t'') at each separate time is a separate integration variable. So it is legitimate to change variables in the integral by shifting: <math> x(t)=u(t)+\epsilon(t)</math> where ε(''t'') is a different shift at each time but ε(0) = ε(''T'') = 0, since the endpoints are not integrated:
| |
| :<math>
| |
| \int \psi_0(x) \int_{u(0)=x} e^{{\rm i}S(u+\epsilon,\dot{u}+\dot{\epsilon})} Du
| |
| \,</math>
| |
| | |
| The change in the integral from the shift is, to first infinitesimal order in epsilon:
| |
| :<math>
| |
| \int \psi_0(x) \int_{u(0)=x} \left( \int {\partial S \over \partial u } \epsilon + { \partial S \over \partial \dot{u} } \dot{\epsilon} dt \right) e^{iS} Du
| |
| \,</math>
| |
| | |
| which, integrating by parts in ''t'', gives:
| |
| :<math>
| |
| \int \psi_0(x) \int_{u(0)=x} -\left( \int \left({d\over dt} {\partial S\over \partial \dot{u}} - {\partial S \over \partial u}\right)\epsilon(t) dt \right) e^{iS} Du
| |
| \,</math>
| |
| | |
| But this was just a shift of integration variables, which doesn't change the value of the integral for any choice of ε(''t''). The conclusion is that this first order variation is zero for an arbitrary initial state and at any arbitrary point in time:
| |
| :<math>
| |
| \langle \psi_0| {\delta S \over \delta x}(t) |\psi_0 \rangle = 0
| |
| \,</math>
| |
| this is the Heisenberg equations of motion.
| |
| | |
| If the action contains terms which multiply ''ẋ'' and ''x'', at the same moment in time, the manipulations above are only heuristic, because the multiplication rules for these quantities is just as noncommuting in the path integral as it is in the operator formalism.
| |
| | |
| === Stationary phase approximation ===
| |
| | |
| If the variation in the action exceeds ''ħ'' by many orders of magnitude, we typically have destructive phase interference other than in the vicinity of those trajectories satisfying the [[Euler–Lagrange equation]], which is now reinterpreted as the condition for constructive phase interference.
| |
| | |
| === Canonical commutation relations ===
| |
| | |
| The formulation of the path integral does not make it clear at first sight that the quantities x and p do not commute. In the path integral, these are just integration variables and they have no obvious ordering. Feynman discovered that the non-commutativity is still there.<ref>{{cite journal|author= Feynman, R. P. | title=Space-Time Approach to Non-Relativistic Quantum Mechanics |journal="Reviews of Modern Physics"|volume= 20| pages=367–387|year=1948|doi= 10.1103/RevModPhys.20.367 |bibcode = 1948RvMP...20..367F|issue= 2 }}</ref>
| |
| | |
| To see this, consider the simplest path integral, the brownian walk. This is not yet quantum mechanics, so in the path-integral the action is not multiplied by ''i'':
| |
| | |
| :<math>
| |
| S= \int \left( {dx \over dt} \right)^2 dt
| |
| \,</math>
| |
| | |
| The quantity ''x''(''t'') is fluctuating, and the derivative is defined as the limit of a discrete difference.
| |
| | |
| :<math>
| |
| {dx \over dt} = {x(t+\epsilon) - x(t) \over \epsilon}
| |
| \,</math>
| |
| | |
| Note that the distance that a random walk moves is proportional to √''t'', so that:
| |
| :<math>
| |
| x(t+\epsilon) - x(t) \approx \sqrt{\epsilon}
| |
| \,</math>
| |
| This shows that the random walk is not differentiable, since the ratio that defines the derivative diverges with probability one.
| |
| | |
| The quantity ''x ẋ'' is ambiguous, with two possible meanings:
| |
| | |
| :<math>[1] = x { dx\over dt} = x(t) {(x(t+\epsilon) - x(t)) \over \epsilon } \,</math>
| |
| | |
| :<math>[2] = x {dx \over dt} = x(t+\epsilon) {(x(t+\epsilon) - x(t)) \over \epsilon} \,</math>
| |
| | |
| In elementary calculus, the two are only different by an amount which goes to zero as ε goes to zero. But in this case, the difference between the two is not zero:
| |
| | |
| :<math>
| |
| [2] - [1] = {( x(t + \epsilon) - x(t) )^2 \over \epsilon} \approx {\epsilon \over \epsilon}
| |
| \,</math>
| |
| | |
| give a name to the value of the difference for any one random walk:
| |
| ::<math>
| |
| {(x(t+\epsilon)- x(t))^2 \over \epsilon} = f(t)
| |
| \,</math>
| |
| | |
| and note that ''f''(''t'') is a rapidly fluctuating statistical quantity, whose average value is 1, i.e. a normalized "Gaussian process". The fluctuations of such a quantity can be described by a statistical Lagrangian
| |
| :<math>\mathcal L = (f(t)-1)^2 \,,</math>
| |
| and the equations of motion for ''f'' derived from extremizing the action ''S'' corresponding to <math> \mathcal L</math> just set it equal to 1. In physics, such a quantity is "equal to 1 as an [[identity operator]]". In mathematics, it "weakly converges to 1". In either case, it is 1 in any expectation value, or when averaged over any interval, or for all practical purpose.
| |
| | |
| Defining the time order to ''be'' the operator order:
| |
| ::<math>
| |
| [x, \dot x] = x {dx\over dt} - {dx \over dt} x = 1
| |
| \,</math>
| |
| | |
| This is called the [[Itō lemma]] in [[stochastic calculus]], and the (euclideanized) canonical commutation relations in physics.
| |
| | |
| For a general statistical action, a similar argument shows that
| |
| :<math>
| |
| \left[x , {\partial S \over \partial \dot x} \right] = 1
| |
| \,</math>
| |
| and in quantum mechanics, the extra imaginary unit in the action converts this to the canonical commutation relation,
| |
| :<math>
| |
| [x,p ] ={\rm i}
| |
| \,</math>
| |
| | |
| === Particle in curved space ===
| |
| | |
| For a particle in curved space the kinetic term depends on the position and the above time slicing cannot be
| |
| applied, this being a manifestation of the notorious [[operator ordering problem]] in Schrödinger quantum mechanics. One may, however, solve this problem by transforming the time-sliced flat-space path integral to curved space using a multivalued coordinate transformation ([[nonholonomic mapping]] explained [http://www.physik.fu-berlin.de/~kleinert/b5/psfiles/pthic10.pdf here]).
| |
| | |
| ===The path integral and the partition function===
| |
| | |
| The path integral is just the generalization of the integral above to all quantum mechanical problems—
| |
| :<math>Z = \int Dx\, e^{{\rm i}\mathcal{S}[x]/\hbar}</math> where <math>\mathcal{S}[x]=\int_0^T \mathrm{d}t L[x(t)]</math>
| |
| is the [[action (physics)|action]] of the classical problem in which one investigates the path starting at time ''t=0'' and ending at time ''t'' = ''T'', and ''Dx'' denotes integration over all paths. In the classical limit, <math>\mathcal{S}[x] \gg \hbar</math>, the path of minimum action dominates the integral, because the phase of any path away from this fluctuates rapidly and different contributions cancel.<ref name="Feynman-Hibbs">{{Cite book | last1 = Feynman | first1 = Richard P. (Richard Phillips) | authorlink1 = Richard Feynman | last2 = Hibbs | first2 = Albert R. | authorlink2 = Albert Hibbs | last3 = Styer | first3 = Daniel F. | authorlink3 = Daniel F. Styer | title = Quantum Mechanics and Path Integrals | year = 2010 | publisher = Dover Publications | location = Mineola, N.Y. | isbn = 0-486-47722-3 | pages = 29–31 }}</ref>
| |
| | |
| The connection with [[statistical mechanics]] follows. Considering only paths which begin and end in the same configuration, perform the [[Wick rotation]] ''t'' → ''it'', i.e., make time imaginary, and integrate over all possible beginning/ending configurations. The path integral now resembles the [[partition function (statistical mechanics)|partition function]] of statistical mechanics defined in a [[canonical ensemble]] with temperature <math>1/T\hbar</math>. Strictly speaking, though, this is the partition function for a [[statistical field theory]].
| |
| | |
| Clearly, such a deep analogy between [[quantum mechanics]] and statistical mechanics cannot be dependent on the formulation. In the canonical formulation, one sees that the unitary evolution operator of a state is given by
| |
| :<math>|\alpha;t\rangle=e^{-{\rm i}Ht / \hbar}|\alpha;0\rangle</math>
| |
| where the state α is evolved from time ''t'' = 0. If one makes a Wick rotation here, and finds the amplitude to go from any state, back to the same state in (imaginary) time ''iT'' is given by
| |
| :<math>Z={\rm Tr} [e^{-HT / \hbar}]</math>
| |
| which is precisely the partition function of statistical mechanics for the same system at temperature quoted earlier. One aspect of this equivalence was also known to [[Schrödinger]] who remarked that the equation named after him looked like the [[diffusion equation]] after Wick rotation.
| |
| | |
| === Measure theoretic factors ===
| |
| Sometimes (e.g. a particle moving in curved space) we also have measure-theoretic factors in the functional integral.
| |
| :<math>\int \mu[x] e^{iS[x]} \mathcal{D}x</math>
| |
| This factor is needed to restore unitarity.
| |
| | |
| For instance, if
| |
| :<math>S=\int \left[ \frac{m}{2}g_{ij}\dot{x}^i\dot{x}^j - V(x) \right] dt</math>,
| |
| then it means that each spatial slice is multiplied by the measure √''g''. This measure can't be expressed as a functional multiplying the <math>\mathcal{D}x</math> measure because they belong to entirely different classes.
| |
| | |
| == Quantum field theory ==
| |
| {{Quantum field theory}}
| |
| The path integral formulation was very important for the development of [[quantum field theory]]. Both the Schrödinger and Heisenberg approaches to quantum mechanics single out time, and are not in the spirit of relativity. For example, the Heisenberg approach requires that scalar field operators obey the commutation relation
| |
| | |
| :<math>
| |
| [\phi(x),\partial_t \phi(y) ] = {\rm i} \delta^3(x-y) \,
| |
| </math>
| |
| | |
| for x and y two simultaneous spatial positions, and this is not a relativistically invariant concept. The results of a calculation are covariant at the end of the day, but the symmetry is not apparent in intermediate stages. If naive field theory calculations did not produce infinite answers in the continuum limit, this would not have been such a big problem – it would just have been a bad choice of coordinates. But the lack of symmetry means that the infinite quantities must be cut off, and the bad coordinates make it nearly impossible to cut off the theory without spoiling the symmetry. This makes it difficult to extract the physical predictions, which require a [[renormalization|careful limiting procedure]].
| |
| | |
| The problem of lost symmetry also appears in classical mechanics, where the Hamiltonian formulation also superficially singles out time. The Lagrangian formulation makes the relativistic invariance apparent. In the same way, the path integral is manifestly relativistic. It reproduces the Schrödinger equation, the Heisenberg equations of motion, and the canonical commutation relations and shows that they are compatible with relativity. It extends the Heisenberg type operator algebra to [[operator product expansion|operator product rules]] which are new relations difficult to see in the old formalism.
| |
| | |
| Further, different choices of canonical variables lead to very different seeming formulations of the same theory. The transformations between the variables can be very complicated, but the path integral makes them into reasonably straightforward changes of integration variables. For these reasons, the Feynman path integral has made earlier formalisms largely obsolete.
| |
| | |
| The price of a path integral representation is that the unitarity of a theory is no longer self-evident, but it can be proven by changing variables to some canonical representation. The path integral itself also deals with larger mathematical spaces than is usual, which requires more careful mathematics not all of which has been fully worked out. The path integral historically was not immediately accepted, partly because it took many years to incorporate fermions properly. This required physicists to invent an entirely new mathematical object – the [[Grassmann variable]] – which also allowed changes of variables to be done naturally, as well as allowing [[Faddeev–Popov ghost|constrained quantization]].
| |
| | |
| The integration variables in the path integral are subtly non-commuting. The value of the product of two field operators at what looks like the same point depends on how the two points are ordered in space and time. This makes some naive identities [[Anomaly (physics)|fail]].
| |
| | |
| === The propagator ===
| |
| | |
| In relativistic theories, there is both a particle and field representation for every theory. The field representation is a sum over all field configurations, and the particle representation is a sum over different particle paths.
| |
| | |
| The nonrelativistic formulation is traditionally given in terms of particle paths, not fields. There, the path integral in the usual variables, with fixed boundary conditions, gives the probability amplitude for a particle to go from point ''x'' to point ''y'' in time ''T''.
| |
| | |
| :<math>
| |
| K(x,y;T) = \langle y;T|x;0 \rangle = \int_{x(0)=x}^{x(T)=y} e^{i S[x]} Dx
| |
| \,</math>
| |
| | |
| This is called the [[propagator]]. Superposing different values of the initial position <math>x</math> with an arbitrary initial state <math>\psi_0(x)</math> constructs the final state.
| |
| :<math>
| |
| \psi_T(y) = \int_{x} \psi_0(x) K(x,y;T) dx = \int^{x(T)=y} \psi_0(x(0)) e^{i S[x]} Dx
| |
| \,</math>
| |
| | |
| For a spatially homogenous system, where ''K''(''x, y'') is only a function of (''x − y''), the integral is a [[convolution]], the final state is the initial state convolved with the propagator.
| |
| :<math>
| |
| \psi_T = \psi_0 * K(;T)
| |
| \,</math>
| |
| | |
| For a free particle of mass m, the propagator can be evaluated either explicitly from the path integral or by noting that the Schrödinger equation is a diffusion equation in imaginary time and the solution must be a normalized Gaussian:
| |
| :<math>
| |
| K(x,y;T) \propto e^{i m(x-y)^2\over 2T}
| |
| </math>
| |
| | |
| Taking the Fourier transform in (''x − y'') produces another Gaussian:
| |
| | |
| :<math>
| |
| K(p;T) = e^{i T p^2\over 2m}
| |
| </math>
| |
| | |
| and in ''p''-space the proportionality factor here is constant in time, as will be verified in a moment. The Fourier transform in time, extending ''K''(''p; T'') to be zero for negative times, gives the Green's Function, or the frequency space propagator:
| |
| :<math>
| |
| G_F(p,E) = {-i \over E - {\vec{p}^2\over 2m} + i\epsilon}
| |
| </math>
| |
| | |
| Which is the reciprocal of the operator which annihilates the wavefunction in the Schrödinger equation, which wouldn't have come out right if the proportionality factor weren't constant in the ''p''-space representation.
| |
| | |
| The infinitesimal term in the denominator is a small positive number which guarantees that the inverse Fourier transform in E will be nonzero only for future times. For past times, the inverse Fourier transform contour closes toward values of E where there is no singularity. This guarantees that K propagates the particle into the future and is the reason for the subscript on G. The infinitesimal term can be interpreted as an infinitesimal rotation toward imaginary time.
| |
| | |
| It is also possible to reexpress the nonrelativistic time evolution in terms of propagators which go toward the past, since the Schrödinger equation is time-reversible. The past propagator is the same as the future propagator except for the obvious difference that it vanishes in the future, and in the gaussian ''t'' is replaced by (−''t''). In this case, the interpretation is that these are the quantities to convolve the final wavefunction so as to get the initial wavefunction.
| |
| :<math>
| |
| G_B(p,E) = { - i \over - E - {i\vec{p}^2\over 2m} + i\epsilon}
| |
| </math>
| |
| Given the nearly identical only change is the sign of ''E'' and ε. The parameter ''E'' in the Green's function can either be the energy if the paths are going toward the future, or the negative of the energy if the paths are going toward the past.
| |
| | |
| For a nonrelativistic theory, the time as measured along the path of a moving particle and the time as measured by an outside observer are the same. In relativity, this is no longer true. For a relativistic theory the propagator should be defined as the sum over all paths which travel between two points in a fixed proper time, as measured along the path. These paths describe the trajectory of a particle in space and in time.
| |
| | |
| :<math>
| |
| K(x-y,\Tau) = \int_{x(0)=x}^{x(\Tau)=y} e^{i \int_0^\Tau \sqrt{{\dot x}^2} - \alpha d\tau}
| |
| \,</math>
| |
| | |
| The integral above is not trivial to interpret, because of the square root. Fortunately, there is a heuristic trick. The sum is over the relativistic arclength of the path of an oscillating quantity, and like the nonrelativistic path integral should be interpreted as slightly rotated into imaginary time. The function <math>K(x-y,\tau)</math> can be evaluated when the sum is over paths in Euclidean space.
| |
| :<math>
| |
| K(x-y,\Tau) = e^{-\alpha \Tau} \int_{x(0)=x}^{x(\Tau)=y} e^{-L}
| |
| \,</math>
| |
| | |
| This describes a sum over all paths of length <math>\Tau</math> of the exponential of minus the length. This can be given a probability interpretation. The sum over all paths is a probability average over a path constructed step by step. The total number of steps is proportional to <math>\Tau</math>, and each step is less likely the longer it is. By the [[central limit theorem]], the result of many independent steps is a Gaussian of variance proportional to <math>\Tau</math>.
| |
| :<math>
| |
| K(x-y,\Tau) = e^{-\alpha \Tau} e^{-(x-y)^2\over \Tau}
| |
| \,</math>
| |
| | |
| The usual definition of the relativistic propagator only asks for the amplitude is to travel from ''x'' to ''y'', after summing over all the possible proper times it could take.
| |
| :<math>
| |
| K(x-y) = \int_0^{\infty} K(x-y,\Tau) W(\Tau) d\Tau
| |
| \,</math>
| |
| Where <math>W(\Tau)</math> is a weight factor, the relative importance of paths of different proper time. By the translation symmetry in proper time, this weight can only be an exponential factor, and can be absorbed into the constant α.
| |
| :<math>
| |
| K(x-y) = \int_0^{\infty} e^{-{(x-y)^2\over\Tau} -\alpha \Tau} d\Tau
| |
| \,</math>
| |
| | |
| This is the [[Feynman_diagram#Schwinger_representation|Schwinger representation]]. Taking a Fourier transform over the variable (''x − y'') can be done for each value of <math>\Tau</math> separately, and because each separate <math>\Tau</math> contribution is a Gaussian, gives whose fourier transform is another Gaussian with reciprocal width. So in ''p''-space, the propagator can be reexpressed simply:
| |
| :<math>
| |
| K(p) = \int_0^{\infty} e^{-\Tau p^2 - \Tau \alpha} d\Tau = {1\over p^2 + \alpha }
| |
| \,</math>
| |
| | |
| Which is the Euclidian propagator for a scalar particle. Rotating ''p''<sub>0</sub> to be imaginary gives the usual relativistic propagator, up to a factor of (−''i'') and an ambiguity which will be clarified below.
| |
| :<math>
| |
| K(p) = {i\over p_0^2 - \vec{p}^2 - m^2}
| |
| \,</math>
| |
| | |
| This expression can be interpreted in the nonrelativistic limit, where it is convenient to split it by [[partial fractions]]:
| |
| | |
| :<math>
| |
| 2 p_0 K(p) = {i \over p_0 - \sqrt{\vec{p}^2 + m^2}} + {i \over p_0 + \sqrt{\vec{p}^2 + m^2}}
| |
| </math>
| |
| | |
| For states where one nonrelativistic particle is present, the initial wavefunction has a frequency distribution concentrated near ''p''<sub>0</sub> = ''m''. When convolving with the propagator, which in ''p'' space just means multiplying by the propagator, the second term is suppressed and the first term is enhanced. For frequencies near ''p''<sub>0</sub> = ''m'', the dominant first term has the form:
| |
| | |
| :<math>
| |
| 2m K_\mathrm{NR}(p) = {i \over (p_0-m) - {\vec{p}^2\over 2m} }
| |
| </math>
| |
| | |
| This is the expression for the nonrelativistic [[Green's function]] of a free Schrödinger particle.
| |
| | |
| The second term has a nonrelativistic limit also, but this limit is concentrated on frequencies which are negative. The second pole is dominated by contributions from paths where the proper time and the coordinate time are ticking in an opposite sense, which means that the second term is to be interpreted as the antiparticle. The nonrelativistic analysis shows that with this form the antiparticle still has positive energy.
| |
| | |
| The proper way to express this mathematically is that, adding a small suppression factor in proper time, the limit where ''t'' → −∞ of the first term must vanish, while the ''t'' → +∞ limit of the second term must vanish. In the fourier transform, this means shifting the pole in ''p''<sub>0</sub> slightly, so that the inverse fourier transform will pick up a small decay factor in one of the time directions:
| |
| | |
| :<math>
| |
| K(p) = {i \over p_0 - \sqrt{\vec{p}^2 + m^2} + i\epsilon} + {i \over p_0 - \sqrt{\vec{p}^2+m^2} - i\epsilon}
| |
| </math>
| |
| | |
| Without these terms, the pole contribution could not be unambiguously evaluated when taking the inverse Fourier transform of ''p''<sub>0</sub>. The terms can be recombined:
| |
| | |
| :<math>
| |
| K(p) = { i \over {p^2 - m^2 + i\epsilon}}
| |
| </math>
| |
| | |
| Which when factored, produces opposite sign infinitesimal terms in each factor. This is the mathematically precise form of the relativistic particle propagator, free of any ambiguities. The ε term introduces a small imaginary part to the α = ''m''<sup>2</sup>, which in the Minkowski version is a small exponential suppression of long paths.
| |
| | |
| So in the relativistic case, the Feynman path-integral representation of the propagator includes paths which go backwards in time, which describe antiparticles. The paths which contribute to the relativistic propagator go forward and backwards in time, and the [[Feynman–Stueckelberg interpretation|interpretation]] of this is that the amplitude for a free particle to travel between two points includes amplitudes for the particle to fluctuate into an antiparticle, travel back in time, then forward again.
| |
| | |
| Unlike the nonrelativistic case, it is impossible to produce a relativistic theory of local particle propagation without including antiparticles. All local differential operators have inverses which are nonzero outside the lightcone, meaning that it is impossible to keep a particle from travelling faster than light. Such a particle cannot have a Greens function which is only nonzero in the future in a relativistically invariant theory.
| |
| | |
| === Functionals of fields ===
| |
| | |
| However, the path integral formulation is also extremely important in ''direct'' application to quantum field theory, in which the "paths" or histories being considered are not the motions of a single particle, but the possible time evolutions of a [[field (physics)|field]] over all space. The action is referred to technically as a [[functional (mathematics)|functional]] of the field: ''S''[ϕ] where the field ϕ(''x''<sup>μ</sup>) is itself a function of space and time, and the '''square brackets''' are a '''reminder''' that the action depends on '''all''' the '''field's values everywhere''', not just some particular value. In principle, one integrates Feynman's amplitude over the class of all possible combinations of values that the field could have anywhere in [[space–time]].
| |
| | |
| Much of the formal study of QFT is devoted to the properties of the resulting functional integral, and much effort (not yet entirely successful) has been made toward making these [[functional integral]]s mathematically precise.
| |
| | |
| Such a functional integral is extremely similar to the [[partition function (statistical mechanics)|partition function]] in [[statistical mechanics]]. Indeed, it is sometimes ''called'' a [[partition function (quantum field theory)|partition function]], and the two are essentially mathematically identical except for the factor of ''i'' in the exponent in Feynman's postulate 3. [[Analytic continuation|Analytically continuing]] the integral to an imaginary time variable (called a [[Wick rotation]]) makes the functional integral even more like a statistical partition function, and also tames some of the mathematical difficulties of working with these integrals.
| |
| | |
| === Expectation values ===
| |
| | |
| In [[quantum field theory]], if the [[action (physics)|action]] is given by the [[functional (mathematics)|functional]] <math>\mathcal{S}</math> of field configurations (which only depends locally on the fields), then the [[time ordered]] [[vacuum expectation value]] of [[polynomially bounded]] functional ''F'', <''F''>, is given by
| |
| | |
| :<math>\left\langle F\right\rangle=\frac{\int \mathcal{D}\phi F[\phi]e^{i\mathcal{S}[\phi]}}{\int\mathcal{D}\phi e^{i\mathcal{S}[\phi]}}</math>
| |
| | |
| The symbol <math>\int \mathcal{D}\phi</math> here is a concise way to represent the infinite-dimensional integral over all possible field configurations on all of space–time. As stated above, we put the unadorned path integral in the denominator to normalize everything properly.
| |
| | |
| === As a probability ===
| |
| Strictly speaking the only question that can be asked in physics is: ''"What fraction of states satisfying condition A also satisfy condition B?"'' The answer to this is a number between 0 and 1 which can be interpreted as a [[probability]] which is written as P(B|A). In terms of path integration, since <math>P(B|A) = \frac{P(A \cap B)}{P(A)}</math> this means:
| |
| | |
| :<math>P(B|A) = \frac{\sum_{F\subset A \cap B}\left| \int \mathcal{D}\phi O_{in}[\phi]e^{i\mathcal{S}[\phi]} F[\phi]\right|^2}{\sum_{F\subset A} \left|\int\mathcal{D}\phi O_{in}[\phi] e^{i\mathcal{S}[\phi]} F[\phi]\right|^2}</math>
| |
| | |
| where the functional ''O''<sub>in</sub>[ϕ] is the superposition of all incoming states that could lead to the states we are interested in. In particular this could be a state corresponding to the state of the Universe just after the [[big bang]] although for actual calculation this can be simplified using heuristic methods. Since this expression is a quotient of path integrals it is naturally normalised.
| |
| | |
| === Schwinger–Dyson equations ===
| |
| {{Main|Schwinger–Dyson equation}}
| |
| | |
| Since this formulation of quantum mechanics is analogous to classical action principles, one might expect that identities concerning the action in classical mechanics would have quantum counterparts derivable from a functional integral. This is often the case.
| |
| | |
| In the language of functional analysis, we can write the [[Euler–Lagrange equation]]s as
| |
| :<math>\frac{\delta \mathcal{S}[\phi]}{\delta \phi}=0</math>
| |
| (the left-hand side is a [[functional derivative]]; the equation means that the action is stationary under small changes in the field configuration). The quantum analogues of these equations are called the [[Schwinger–Dyson equation]]s.
| |
| | |
| If the [[functional measure]] <math>\mathcal{D}\phi</math> turns out to be [[translationally invariant]] (we'll assume this for the rest of this article, although this does not hold for, let's say [[nonlinear sigma model]]s) and if we assume that after a [[Wick rotation]]
| |
| | |
| :<math>e^{i\mathcal{S}[\phi]},</math>
| |
| | |
| which now becomes
| |
| | |
| :<math>e^{-H[\phi]}\,</math>
| |
| | |
| for some ''H'', goes to zero faster than a [[Multiplicative inverse|reciprocal]] of any [[polynomial]] for large values of φ, we can [[integration by parts|integrate by parts]] (after a Wick rotation, followed by a Wick rotation back) to get the following Schwinger–Dyson equations for the expectation:
| |
| | |
| :<math>\left\langle \frac{\delta F[\phi]}{\delta \phi} \right\rangle = -i \left\langle F[\phi]\frac{\delta \mathcal{S}[\phi]}{\delta\phi} \right\rangle</math>
| |
| | |
| for any polynomially bounded functional ''F''.
| |
| | |
| :<math>\left\langle F_{,i} \right\rangle = -i \left\langle F \mathcal{S}_{,i} \right\rangle</math>
| |
| | |
| in the [[deWitt notation]].
| |
| | |
| These equations are the analog of the [[on shell]] EL equations.
| |
| | |
| If ''J'' (called the [[source field]]) is an element of the [[dual space]] of the field configurations (which has at least an [[affine structure]] because of the assumption of the [[translational invariance]] for the functional measure), then, the [[generating functional]] ''Z'' of the source fields is '''defined''' to be:
| |
| | |
| :<math>Z[J]=\int \mathcal{D}\phi e^{i(\mathcal{S}[\phi] + \left\langle J,\phi \right\rangle)}.</math>
| |
| | |
| Note that
| |
| | |
| :<math>\frac{\delta^n Z}{\delta J(x_1) \cdots \delta J(x_n)}[J] = i^n \, Z[J] \, {\left\langle \phi(x_1)\cdots \phi(x_n)\right\rangle}_J</math>
| |
| | |
| or
| |
| | |
| :<math>Z^{,i_1\dots i_n}[J]=i^n Z[J] {\left \langle \phi^{i_1}\cdots \phi^{i_n}\right\rangle}_J</math>
| |
| | |
| where
| |
| | |
| :<math>{\left\langle F \right\rangle}_J=\frac{\int \mathcal{D}\phi F[\phi]e^{i(\mathcal{S}[\phi] + \left\langle J,\phi \right\rangle)}}{\int\mathcal{D}\phi e^{i(\mathcal{S}[\phi] + \left\langle J,\phi \right\rangle)}}.</math>
| |
| | |
| Basically, if <math>\mathcal{D}\phi e^{i\mathcal{S}[\phi]}</math> is viewed as a functional distribution (this shouldn't be taken too literally as an interpretation of [[Quantum field theory|QFT]], unlike its Wick rotated [[statistical mechanics]] analogue, because we have [[time ordering]] complications here!), then <math>\left\langle\phi(x_1)\cdots \phi(x_n)\right\rangle</math> are its [[moment (mathematics)|moments]] and Z is its [[Fourier transform]].
| |
| | |
| If ''F'' is a functional of φ, then for an [[Operator (mathematics)|operator]] ''K'', ''F''[''K''] is defined to be the operator which substitutes ''K'' for φ. For example, if
| |
| | |
| :<math>F[\phi]=\frac{\partial^{k_1}}{\partial x_1^{k_1}}\phi(x_1)\cdots \frac{\partial^{k_n}}{\partial x_n^{k_n}}\phi(x_n)</math>
| |
| | |
| and ''G'' is a functional of ''J'', then
| |
| | |
| :<math>F\left[-i\frac{\delta}{\delta J}\right] G[J] = (-i)^n \frac{\partial^{k_1}}{\partial x_1^{k_1}}\frac{\delta}{\delta J(x_1)} \cdots \frac{\partial^{k_n}}{\partial x_n^{k_n}}\frac{\delta}{\delta J(x_n)} G[J].</math>
| |
| | |
| Then, from the properties of the [[functional integral]]s
| |
| | |
| :<math>{\left \langle \frac{\delta \mathcal{S}}{\delta \phi(x)}\left[\phi \right]+J(x)\right\rangle}_J=0</math>
| |
| | |
| we get the "master" Schwinger–Dyson equation:
| |
| | |
| :<math>\frac{\delta \mathcal{S}}{\delta \phi(x)}\left[-i \frac{\delta}{\delta J}\right]Z[J]+J(x)Z[J]=0</math>
| |
| | |
| or
| |
| | |
| :<math>\mathcal{S}_{,i}[-i\partial]Z+J_i Z=0.</math>
| |
| | |
| If the functional measure is not translationally invariant, it might be possible to express it as the product <math>M\left[\phi\right]\,\mathcal{D}\phi</math> where ''M'' is a functional and <math>\mathcal{D}\phi</math> is a translationally invariant measure. This is true, for example, for nonlinear sigma models where the [[target space]] is diffeomorphic to '''R'''<sup>''n''</sup>. However, if the [[target manifold]] is some topologically nontrivial space, the concept of a translation does not even make any sense.
| |
| | |
| In that case, we would have to replace the <math>\mathcal{S}</math> in this equation by another functional <math>\hat{\mathcal{S}}=\mathcal{S}-i\ln(M)</math>
| |
| | |
| If we expand this equation as a [[Taylor series]] about ''J'' = 0, we get the entire set of Schwinger–Dyson equations.
| |
| | |
| == Localization ==
| |
| The path integrals are usually thought of as being the sum of all paths through an infinite space–time. However, in [[Local quantum field theory]] we would restrict everything to lie within a finite ''causally complete'' region, for example inside a double light-cone. This gives a more mathematically precise and physically rigorous definition of quantum field theory.
| |
| | |
| == Functional identity ==
| |
| | |
| If we perform a [[Wick rotation]] inside the functional integral, professors J. Garcia and [[Gerard 't Hooft]] showed using a functional differential equation, that
| |
| | |
| <math> \int D[x]e^{-\mathcal{S}[x]/\hbar}=-A[x]\sum_{n=0}^{\infty}(\hbar)^{n+1}\delta^{n} e^{-J/\hbar} \text{,}</math>
| |
| | |
| where ''S'' is the Wick-rotated classical action of the particle, ''J'' is the classical action with an extra term "''x''", delta (here) is the [[functional derivative operator]] and
| |
| | |
| :<math> A[x]=\exp\left(\frac{1}{\hbar}\int X(t)\,\mathrm{d}t\right) \text{.}</math>
| |
| | |
| === Ward–Takahashi identities ===
| |
| :''See main article [[Ward–Takahashi identity]]''.
| |
| | |
| Now how about the [[on shell]] [[Noether's theorem]] for the classical case? Does it have a quantum analog as well? Yes, but with a caveat. The functional measure would have to be invariant under the one parameter group of symmetry transformation as well.
| |
| | |
| Let's just assume for simplicity here that the symmetry in question is local (not local in the sense of a [[gauge symmetry]], but in the sense that the transformed value of the field at any given point under an infinitesimal transformation would only depend on the field configuration over an arbitrarily small neighborhood of the point in question). Let's also assume that the action is local in the sense that it is the integral over spacetime of a [[Lagrangian]], and that <math>Q[\mathcal{L}(x)]=\partial_\mu f^\mu (x)</math> for some function f where f only depends locally on φ (and possibly the spacetime position).
| |
| | |
| If we don't assume any special boundary conditions, this would not be a "true" symmetry in the true sense of the term in general unless f=0 or something. Here, Q is a [[Derivation (abstract algebra)|derivation]] which generates the one parameter group in question. We could have [[antiderivation]]s as well, such as [[BRST quantization|BRST]] and [[supersymmetry]].
| |
| | |
| Let's also assume <math>\int \mathcal{D}\phi Q[F][\phi]=0</math> for any polynomially bounded functional F. This property is called the invariance of the measure. And this does not hold in general. See [[anomaly (physics)]] for more details.
| |
| | |
| Then,
| |
| | |
| :<math>\int \mathcal{D}\phi\, Q\left[F e^{iS}\right][\phi]=0,</math>
| |
| | |
| which implies
| |
| | |
| :<math>\left\langle Q[F]\right\rangle +i\left\langle F\int_{\partial V} f^\mu ds_\mu\right\rangle=0</math>
| |
| | |
| where the integral is over the boundary. This is the quantum analog of Noether's theorem.
| |
| | |
| Now, let's assume even further that Q is a local integral
| |
| | |
| :<math>Q=\int d^dx q(x)</math>
| |
| | |
| where
| |
| | |
| :<math>q(x)[\phi(y)] = \delta^{(d)}(X-y)Q[\phi(y)] \,</math>
| |
| | |
| so that
| |
| | |
| :<math>q(x)[S]=\partial_\mu j^\mu (x) \,</math>
| |
| | |
| where
| |
| | |
| :<math>j^{\mu}(x)=f^\mu(x)-\frac{\partial}{\partial (\partial_\mu \phi)}\mathcal{L}(x) Q[\phi] \,</math>
| |
| | |
| (this is assuming the Lagrangian only depends on φ and its first partial derivatives! More general Lagrangians would require a modification to this definition!). Note that we're NOT insisting that q(x) is the generator of a symmetry (i.e. we are ''not'' insisting upon the [[gauge principle]]), but just that ''Q'' is. And we also assume the even stronger assumption that the functional measure is locally invariant:
| |
| | |
| :<math>\int \mathcal{D}\phi\, q(x)[F][\phi]=0.</math>
| |
| | |
| Then, we would have
| |
| | |
| :<math>\left\langle q(x)[F] \right\rangle +i\left\langle F q(x)[S]\right\rangle=\left\langle q(x)[F]\right\rangle +i\left\langle F\partial_\mu j^\mu(x)\right\rangle=0.</math>
| |
| | |
| Alternatively,
| |
| | |
| :<math>q(x)[S]\left[-i \frac{\delta}{\delta J}\right]Z[J]+J(x)Q[\phi(x)]\left[-i \frac{\delta}{\delta J}\right]Z[J]=\partial_\mu j^\mu(x)\left[-i \frac{\delta}{\delta J}\right]Z[J]+J(x)Q[\phi(x)]\left[-i \frac{\delta}{\delta J}\right]Z[J]=0.</math>
| |
| | |
| The above two equations are the '''Ward–Takahashi identities'''.
| |
| | |
| Now for the case where f=0, we can forget about all the boundary conditions and locality assumptions. We'd simply have
| |
| | |
| :<math>\left\langle Q[F]\right\rangle =0.</math>
| |
| | |
| Alternatively,
| |
| | |
| :<math>\int d^dx\, J(x)Q[\phi(x)]\left[-i \frac{\delta}{\delta J}\right]Z[J]=0.</math>
| |
| | |
| == The need for regulators and renormalization ==
| |
| | |
| Path integrals as they are defined here require the introduction of [[Regularization (physics)|regulators]]. Changing the scale of the regulator leads to the [[renormalization group]]. In fact, renormalization is the major obstruction to making path integrals well-defined.
| |
| | |
| == The path integral in quantum-mechanical interpretation ==
| |
| | |
| In one philosophical [[interpretation of quantum mechanics]], the "sum over histories" interpretation, the path integral is taken to be fundamental and reality is viewed as a single indistinguishable "class" of paths which all share the same events. For this interpretation, it is crucial to understand what exactly an event is. The sum over histories method gives identical results to canonical quantum mechanics, and Sinha and Sorkin (see the reference below) claim the interpretation explains the [[Einstein–Podolsky–Rosen paradox]] without resorting to [[action at a distance|nonlocality]]. (Note that the Copenhagen/pragmatism interpretation claims there is no paradox—only a sloppy materialism motivated question on the part of EPR—Joseph Wienberg a lecture. On the other hand, the fact that the EPR thought experiment (and its result) does represent the results of a QM experiment says that (despite the path dependence of parallelness/anti-parallelness in curved space) all contributions of paths close to black holes cancel in the action for an EPR style experiment here on earth.)
| |
| | |
| Some advocates of interpretations of quantum mechanics emphasizing [[decoherence]] have attempted to make more rigorous the notion of extracting a classical-like "coarse-grained" history from the space of all possible histories.
| |
| | |
| == Quantum Gravity ==
| |
| Whereas in Quantum Theory the path integral formulation is fully equivalent to other formulations, it may be that it can be extended to quantum gravity, which would make it different from the [[Hilbert space]] model. Feynman had some success in this direction and his work has been extended by [[Hawking]] and others.<ref>"Most of the Good Stuff", ''Memories Of Richard Feynman'', edited by Laurie M. Brown and John S. Rigden, American Institute of Physics, the chapter by Murray Gell-Mann.</ref> The most developed approach is that of [[spinfoam]]s.
| |
| | |
| == See also ==
| |
| *[[Theoretical and experimental justification for the Schrödinger equation]]
| |
| *[[Static forces and virtual-particle exchange]]
| |
| *[[Feynman checkerboard]]
| |
| *[[Propagator]]s
| |
| *[[Wheeler–Feynman absorber theory]]
| |
| *[[Feynman–Kac formula]]
| |
| | |
| == References ==
| |
| {{Reflist}}
| |
| | |
| == Notes ==
| |
| {{Reflist|group="note"}}
| |
| | |
| == Suggested reading ==
| |
| * {{cite book |author=Feynman, R. P. and Hibbs, A. R. |year=1965 |title=Quantum Mechanics and Path Integrals |place=New York |publisher=McGraw-Hill |isbn=0-07-020650-3}} <small>The historical reference, written by the inventor of the path integral formulation himself and one of his students.<small/>
| |
| * {{cite book |author=[[Hagen Kleinert]] |year=2004 |title=Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets |edition=4th |place=Singapore |publisher=World Scientific |isbn=981-238-107-4 |url=http://www.physik.fu-berlin.de/~kleinert/b5}}
| |
| * {{cite book |author=Zinn Justin, Jean |year=2004 |title=Path Integrals in Quantum Mechanics |publisher=Oxford University Press |isbn=0-19-856674-3}} A highly readable introduction to the subject.
| |
| * {{cite book |author=Schulman, Larry S. |year=1981 |title=Techniques & Applications of Path Integration |place=New York |publisher=John Wiley & Sons |isbn=0486445283}} A modern reference on the subject.
| |
| * {{cite book |author=Ahmad, Ishfaq |authorlink=Ishfaq Ahmad |title=Mathematical Integrals in Quantum Nature |series=The Nucleus |year=1971 |pages=189–209}}
| |
| * {{cite book |author=Grosche, Christian & Steiner, Frank |year=1998 |title=Handbook of Feynman Path Integrals |series=Springer Tracts in Modern Physics 145 |publisher=Springer-Verlag |isbn=3-540-57135-3}}
| |
| * {{cite book |author=Ryder, Lewis H. |title=Quantum Field Theory |publisher=Cambridge University Press |year=1985 |isbn=0-521-33859-X}} Highly readable textbook; introduction to relativistic QFT for particle physics.
| |
| * {{cite book |author=Rivers, R.J. |title=Path Integrals Methods in Quantum Field Theory |publisher=Cambridge University Press |year=1987 |isbn=0-521-25979-7}}
| |
| * {{cite book |author=Albeverio, S. & Hoegh-Krohn. R. |title=Mathematical Theory of Feynman Path Integral |series=Lecture Notes in Mathematics 523 |publisher=Springer-Verlag |year=1976 |isbn=0-387-07785-5}}
| |
| * {{cite book |author=Glimm, James, and Jaffe, Arthur |title=Quantum Physics: A Functional Integral Point of View |place=New York |publisher=Springer-Verlag |year=1981 |isbn=0-387-90562-6}}
| |
| * {{cite book |author=Gerald W. Johnson and Michel L. Lapidus |title=The Feynman Integral and Feynman's Operational Calculus |series=Oxford Mathematical Monographs |publisher=Oxford University Press |year=2002 |isbn=0-19-851572-3}}
| |
| * {{cite web |author=Etingof, Pavel |title=Geometry and Quantum Field Theory |publisher=MIT OpenCourseWare |year=2002 |url=http://ocw.mit.edu/courses/mathematics/18-238-geometry-and-quantum-field-theory-fall-2002/index.htm}} This course, designed for mathematicians, is a rigorous introduction to perturbative quantum field theory, using the language of functional integrals.
| |
| * {{cite book |last=Zee |first=Anthony |authorlink=Anthony Zee |title=Quantum Field Theory in a Nutshell |edition=Second |date= |publisher=Princeton University Press |location= |isbn=978-0-691-14034-6 }} A great introduction to Path Integrals (Chapter 1) and QFT in general.
| |
| * {{cite arxiv |last=Grosche |first=Christian |title=An Introduction into the Feynman Path Integral |work=Lecture given at the graduate college {{lang|de|‘Quantenfeldtheorie und deren Anwendung in der Elementarteilchen- und Festkörperphysik’}}, Universität Leipzig, 16–26 November 1992 |year=1992 |eprint=hep-th/9302097}}
| |
| * {{cite arxiv |last=MacKenzie |first=Richard |year=2000 |title=Path Integral Methods and Applications |work=Lectures given at Rencontres du Vietnam: VIth Vietnam School of Physics, Vung Tau, Vietnam, 27 December 1999 – 8 January 2000 |eprint=quant-ph/0004090}}
| |
| * {{cite journal |authorlink=Cécile DeWitt-Morette |last=DeWitt-Morette |first=Cécile |title=Feynman's path integral: Definition without limiting procedure |journal=Communication in Mathematical Physics |volume=28 |issue=1 |year=1972 |pages=47–67 |mr=0309456 |doi=10.1007/BF02099371 |bibcode = 1972CMaPh..28...47D}}
| |
| * {{cite journal |first=Sukanya |last=Sinha |first2=Rafael D. |last2=Sorkin |title=A Sum-over-histories Account of an EPR(B) Experiment |journal=Found. of Phys. Lett. |volume=4 |issue=4 |pages=303–335 |year=1991 |doi=10.1007/BF00665892 |url=http://physics.syr.edu/~sorkin/some.papers/63.eprb.ps |bibcode=1991FoPhL...4..303S }}
| |
| * {{cite journal |authorlink=Pierre Cartier (mathematician) |last=Cartier |first=Pierre |last2=DeWitt-Morette |first2=Cécile |title=A new perspective on Functional Integration |journal=Journal of Mathematical Physics |volume=36 |year=1995 |issue=5 |pages=2137–2340 |doi=10.1063/1.531039 |arxiv=funct-an/9602005 |bibcode = 1995JMP....36.2237C }}
| |
| | |
| == External links ==
| |
| * [http://www.scholarpedia.org/article/Path_integral Path integral on Scholarpedia]
| |
| * [http://www.quantumfieldtheory.info/website_Chap18.pdf Path Integrals in Quantum Theories: A Pedagogic 1st Step]
| |
| | |
| {{DEFAULTSORT:Path Integral Formulation}}
| |
| [[Category:Concepts in physics]]
| |
| [[Category:Statistical mechanics]]
| |
| [[Category:Quantum mechanics]]
| |
| [[Category:Quantum field theory]]
| |