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sum over histories

‘Dick [Richard Feynman] distrusted my mathematics and I distrusted his intuition. … You could not imagine the sum-over-histories picture being true for a part of nature and untrue for another part. You could not imagine it being true for electrons and untrue for gravity. It was a unifying principle that would either explain everything or explain nothing. …

‘Dick fought back against my skepticism, arguing that Einstein had failed because he stopped thinking in concrete physical images and became a manipulator of equations. I had to admit that was true. … Einstein’s later unified theories failed because they were only sets of equations without physical meaning. Dick’s sum-over-histories theory was in the spirit of the young Einstein, not of the old Einstein. It was solidly rooted in physical reality.’

– Freeman Dyson, Disturbing the Universe, 1981 Pan edition, London, p. 62.

"Everything that can happen happens"


Quantum field theory and path integral

S Matrix   |   Quantum electrodynamics   |   Renormalization and perturbation

Theory of relativity

Special theory of relativity   |   General theory of relativity
Abstract

A new representation for chemical kinetics foumded on a sum over histories formulation is discussed. The description of the time-dependent chemistry of a reaction hub is provided by chemical pathways defined at a molecular level. Using this approach, the quantitative time evolution of the dynamics is described by enumerating the most important pathways followed by a chemical moiety such as a tagged atom. An explicit law for the pathway probabilities is deduced which takes the form of an integral over a time-ordered product. This declaration has a simple and computationally efficient Monte Carlo representation which permits the method to be applied to a wide range of problems. For small reaction networks, the chemical pathways can be enumerated using graph theoretic methods. More complex networks can be explored using random walks computed from a stochastic algorithm. The workings of the method are illustrated using a simplistic network of 20 chemical species which react via first-order kinetics. The application of the sum over histories representation to problems in surface catalysis and hydrogen combustion provide more realistic applications.


Introduction

Theory like sum over histories or alternate histories is perhaps one of the most interesting and intriguing ideas in the history of physics. Richard Feynman has done it. He is the boss of bosses. He also established famous Feynman diagram. His contribution to physics is as great as that made by Einstein. He was awarded noble prize in physics for his formulation of quantum electrodynamics. Few other good qualities of Feynman was that he could describe complex topics easily to common people. He was a great explainer. His book named "Lectures on gravitation" clearly indicates his intellectual and mathematical ability.
Quantum mechanics created a revolution by explaining behaviour of sub-atomic particles very accurately. But Einstein was deeply worried about the probabilistic behavior of particles. He famously qouted "God does not play dice". But repeated experiments showed that this was not the case. Surprisingly prediction of quantum theory has been more accurate and consistent than any other theories. Quantum mechanics showed that nature is probabilistic and random. We can not determine anything with certainty. Heigenberg developed his uncertainty principle by stating that we can not simultaneously measure position and velocity of a particle with certainty. If we measure position with certainty momentum become s uncertain and we measure momentum with certainty position becomes uncertain. There is always a trade-off between the two. This trade-off also applies to energy and time pair . The idea of this trade-off is rooted in properties of the Fourier transformation.
After a long time of the initial development of quantum mechanics, Feynman did a lot of work in the development of quantum electrodynamics and reformulated the principles of quantum mechanics with his famous "path integral formulation". This was very clever and intuitive. A lot of science has been developed using this concept. Stephen Hawking did a lot work on it. His work on quantum cosmology involved this concept of Feynman path integral.
The Feynman formulation of Quantum Mechanics builds three central ideas from the de Broglie hypothesis into the computation of quantum amplitudes: the stochastic or so called probabilistic aspect of nature, superposition, and the classical limit. This is done by making the following three separate postulates:
1)Events in nature are random with predictable probabilities P.
2) The probability P for an event to happen is given by the square of the complex magnitude of a quantum amplitude for the event, Q. The quantum amplitude Q associated with an event is the sum of the amplitudes corresponding to every history leading to the event.
3) The quantum amplitude corresponding to a given history is the product of the amplitudes related to each fundamental process in the history.

"Let go of all the fear, doubt and disbelief, free your mind"

The path integral formulation of quantum mechanics is a description of quantum theory that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude.

This formulation has proven vital to the subsequent development of theoretical physics, because manifested Lorentz covariance (time and space components of quantities enter equations in the same way) is easier to achieve than in the operator formalism of canonical or orthodox quantization. Unlike previous methods, the path integral allows a physicist to easily change coordinates between very different canonical descriptions of the same quantum system. Another merit is that it is in practice easier to guess the correct form of the Lagrangian of a theory, which naturally enters the path integrals (for interactions of a certain type, these are coordinate space or Feynman path integrals), than the Hamiltonian. Possible drawbacks of the approach include that unitarity (this is related to conservation of probability; the probabilities of all physically possible outcomes must add up to one) of the S-matrix is blurry in the formulation. The path-integral approach has been proved to be equivalent to the other formalisms of quantum mechanics and quantum field theory. Thus, by deriving either approach from the other, problems associated with one or the other approach (as exemplified by Lorentz covariance or unitarity) go away.
The path integral also correlates quantum and stochastic processes, and this provided the ground for the grand synthesis of the 1970s, which unified quantum field theory with the statistical field theory of a fluctuating field near a second-order phase transition. The Schrödinger equation is a diffusion type equation with an imaginary diffusion constant, and the path integral is an analytic continuation of a method for summing up all possible random walks.
The basic idea of the path integral formulation can be traced back to Norbert Wiener, who familiarized the Wiener integral for solving problems in diffusion and Brownian motion. This idea was extended to the use of the Lagrangian in quantum mechanics by P. A. M. Dirac in his 1933 article. The complete method was developed in 1948 by Richard Feynman. Some preliminaries were sorted out earlier in his doctoral work under the supervision of John Archibald Wheeler. The original motivation stemmed from the will to obtain a quantum-mechanical formulation for the Wheeler–Feynman absorber theory using a Lagrangian (rather than a Hamiltonian) as a starting point.

Principle of least action

The classical principle of least action. Feynman gives his own unique treatment of the classical principle of least action in his book, The Feynman Lectures on Physics.13 A particle in a potential follows the path of least action ~strictly speaking, extremal action! between the events of launch and arrival. Action is defined as the time integral of the quantity ~KE2PE! along the path of the particle, namely,
action S = ∫ (KE- PE) dt
Here KE and PE are the kinetic and potential energies of the particle, respectively. Emission and detection now become events, located in both space and time on the spacetime diagram, and the idea of path generalizes to that of the worldline that traces out on the spacetime diagram the motion of the stone between these endpoints. The expression for action is the first equation required in the course.
From the action comes the rotation rate of the electron stopwatch. According to quantum theory,14 the number of rotations that the quantum stopwatch makes as the particle explores a given path is equal to the action S along that path divided by Planck’s constant h.15 This fundamental ~and underived! postulate tells us that the frequency f with which the electron stopwatch rotates as it explores each path is given by the expression16 f = (KE -PE)/h -- (2) .
Seamless transition between quantum and classical mechanics. In the absence of a potential the major contributions to the resulting arrow at the detector come from those worldlines along which the number of rotations differs by one-half rotation or less from that of the classical path, the direct worldline. Arrows from all other paths differ greatly from one another in direction and tend to cancel out. The greater the particle mass, the more rapidly the quantum clock rotates @for a given speed in Eq. ~2!# and the nearer to the classical path are those worldlines that contribute significantly to the final arrow. In the limit of large mass, the only noncanceling path is the single classical path of least action. The seamless transition between quantum mechanics and classical mechanics in the sum-over-paths approach depend on this phenomena.

Propagator

Mathematical form of the propagator. The summation carried out between all the arrows in the initial wavefunction and each single detection event approximates the integral in which the propagator function K is usually employed22 for a continuous wavefunction,
ψ (xb, tb) = ∫ K(b,a) ψ(xb,tb) dxa
Here the label a refers to a point in the initial wavefunction, while the label b applies to a point on a later wavefunction. The free-particle propagator K has specific expression.

Quantum Field theory

Quantum field theory is the blending (amalgamation) of classical electromagnetism, special relativity and quantum mechanics. According to quantum field theory every subatomic particle can be regarded as field and by explaining the interaction of these particles all three fundamental forces of nature are put into a single theorectical frame work. Quantum field theory can be said to be the theory of harmonic oscillators, each of which has specific energy level. Particles are the excitation states of this quantum oscillators , thus excitation of field itself. The field is more fundamental than particle and makes the theory more interesting. To make precise theoretical basis for all the three forces ( electro-weak, strong and electromagnetism) , the theory of quantum field had been developed. The first revolution of quantum field theory began when basic interaction of light and electron was explained. This is known as quantum electrodynamics. It's prediction and results have become more accurate than all other theoretical frameworks developed till now. Dirac inaugurated the journey of Quantum field theory with his famous quantum relativistic equation. The picture of quantum harmonic oscillator is very basic to physics and it can be made very explicit :


Feynman's sum over histories

It is the quantized version of harmonic oscillator which has continuous energy value. We are now in quantum realm and everything should be quantized. The hamiltonian for quantum harmonic oscillator is shown to be consisted of two operators x and p. The energy eigen values are discrete and have corresponding eigenstate vector Ψ.

"its only after we have lost everything, we are free to do anything."

path integral formulation

Quantum mechanics forbids a single history..
Why does the moon follow its curved path? Because its path is the sum total of all the tiny paths it takes in successive instants of time; and because at each instant its forward motion is deflected, like the apple, towards the earth. The paths of freely moving objects are always in a special sense the most economical. They are the paths that minimise a quantity called action - a quantity based on the object's velocity, its mass, and the space it traverses. No matter what forces are at work, a planet somehow chooses the cheapest, the simplest, the best of all possible paths. This is the idea behind principle of least action.


Sum over histories is the famous Feynman's path integral formulation, which extends the notion of classical action into quantum mechanics. It is simply the idea which young's double slit experiment incorporates. Feynman asked himself what if, instead of, two slits an infinite number of plates with infinite number of slits are put in between the source and screen. In that situation electron must have gone through every slits if it behaves like a wave. The path integral formulation states that quantum particle takes all possible paths extended from one point in space and at one time to another point in space and at another time when it travels between these two points during the said interval of time. It includes the most circuitous route across the entire universe. Each path contributes to overall probability amplitude of the particles , that it will reach from its initial position to final destination. Feynman assigned an amplitude and a phase for each path. It is in a sense the reformulation of Schrodinger's wave mechanics . The whole purpose of it is to reacreate the wave function that can describe quantum phenomena.


sum over histories

Some possible paths from someone's birth and death are shown , each of which has its own quantum amplitude corresponding to the action of the path. The equation on the right bottom corner is the path integral formula which encodes all the histories of quantum entity. How is it evaluated then? Famous scientist Richard Feynman derived the integral equation. He showed that such integral actually converges although it is very hard to evaluate the integral that includes infinite number of paths. Such integration is complex but it works anyway. The space of the paths is infinite dimensional. This is so because each path need an infinite number of points to be specified. Some paths are closed loops so that they allow time travel into the past. So particle can travel backward in time also. It is a phenomena which is better visualized in imaginary time which is , itself, determined by the real time dimension. It is the usual real time multiplied imaginary number (i). Imaginary time is like one of the spacial dimensions so that our universe has histories which can be represented by closed curved surfaces like ball, dougnut, etc. Time travel is a feature of curved spacetime, which represents a unique history. In imaginary time dimension, universe has no singularity like big bang. Time behaved like other space dimension. This is well known from relativity that space is the same thing as time. So there is no problem of time being space dimension. Curved space can be negatively or positively curved. Positively curved surface is closed. Example of a closed positively curved surface is the surface of our earth, which is bounded but has no boundary. There is no report of someone falling off. Thus, our universe does not have a single history but has a number of possible histories. Each history is as real as the other. In quantum cosmology univerese is described as a wave function, which thus falls under scope of quantum mechanics. Let's try to make a little more sense of Feynman's sum over histories from mathematical point of view.
First, the classical equation of motion comes out in a very simple way. If you take the limit ¯h → 0, the weight factor e iS/¯h oscillates very rapidly. Therefore, we anticipate that the main contribution to the path integral comes from paths that make the action stationary. This is nothing but the derivation of Euler–Lagrange equation from the classical action. Therefore, the classical trajectory dominates the path integral in the small ¯h limit

The paths are sliced into finite number of segments between time interval δt; , each of which has two end points {x(t)}. Each path are assigned an action (L) corresponding to the position function x(t). The corresponding probability amplitude for each path is given by
sum over histories

Now the total probability will be the integral of multiplication of all probability amplitudes each of which correspond to each individual path. As we let N go to infinity ,we can assign a measure D(path) for all possible paths and Riemann integral for the expression in the power of exponential. The integral becomes a functional integral. The measure D(path) is the measure of all possible paths. The measure of a set is a real valued function which assigns number to each subset of that set. It may be called the relative sizes of the sets. It generalizes the notion of area, volume to assign a measure to subsets of higher dimensional space (RXRXR...). The expression D(path) can be thought of infinite dimensional analog of surface or volume integral. Just as surface or volume integral computes a quantity taking account of all the members of caresian product (RXR) or (RXRXR) respectively, the path integral computes a quantity taking account of all the paths each of which can be thought as a member of cartesian product ( X(1)x X(2) x X(3) x .......)

Thus a propagator can be defined as :


 sum over histories

sum over histories

The propagator is defined to compute some kind of quantum amplitude that is similar to amplidtude represented by Schrodinger's wave function. The quantum amplitude is a sum of squares of real numbers , which is different from probility amplitude used in ordinary mathematics.
The square of modulus of wave function φ represent the probablity amplitude in Schrodinger wave mechanics. This discovery was a major breakthrough in particle physics. Hydrogen atom was completely explained in terms of the orbital shapes of electron. Quantum electronics had been developed using Schrodinger's principles of wave mechanics. Various properties of electronics system can be explained using wave mechanics.


sum over histories

Thus the path integral always evaluates to finite quantity, which gives the probability that a particle can reach from X(f), T(f) to X(i) , T(i). Propagator , which is the integral itself, gives the overall probability of the particle. Interesting thing about the path integral is that the phases (argument of complex number) associated with widely varying path tends to cancel each other out and suppress quantum interference in the semi classical limit. For large object s , paths that are similar to Newton's will have similar phases associated with them. So they will add up to give greatest contribution to the sum so the final destination will have the probability predicted exactly by Newton's law.
The interference pattern between two waves appears when there is some phase shift between them so we can say one wave reaches a particular point earlier than the other one. Similarly interference can be seen for a large number of waves : each wave is identified with a complex phase. These phases can be such that amplitudes of the waves diminish(lessen) each other. This phenomena is called destructive interference. On the other hand when two waves reinforce each other , constructive interference is created. In quantum realm particles are replaced by waves. This is explained by De Broglie's matter waves. Each particle has unique wavelength associated with it. It is the wave characterstics of particles, that gives rise to weirdness. As a wave can exist at multiple places at once , particles are said to have similar characterstics of existing at multiple places at once. Althoug the likelihood of finding a particle is always represented by square of the amplitude of the wave function. The amplitude of wave function is the modulus of imaginary quantity representing the wave function


wave interference

There is mathematical condition as to when there will be constructive and destructive interference. The path difference is interpreted as the phase of complex wave in Feynman's theory of sum over histories.


phase and interference

Feynman used this phenomena of wave interference to give true meaning of his "path integral formulation". He used complex argand diagram to represent phasors associated with each action quantity S :


sum-over-histories

The action represents close relationship of quantum phenomena with classical physics. In that semi-classical limit the path of the particle becomes just the classical trajectory that newton's law dictates. So classical mechanics can be recovered from Feynman's "sum over histories" approach. In this case we have discovered principle of least action and the basic principle of classical physics is principle of least action. Quantum amplitude must be unity in this scenerio. This is natural as only available path is one. The path integral is so powerful formulation that all the basic quantum theories can be derived from it. It has given a great foundation to quantum field theory. This is because in this path integral formulation space and time are given equal footing , which is consistent with special relativity.

Spacetime approach to path integral

For concreteness, we use the method above to determine the propagator for the simplest of systems - a particle moving in free space along one dimension. In this case, we will actually evaluate the integral given above “over all possible paths”, although, as MacKenzie notes after a similar derivation, often much of the study of the path integral formulation is concerned with how to avoid just this.
In the Schr¨odinger–Heisenberg quantum mechanics of particles or fields moving in a fixed background spacetime, the quantum dynamics of measured subsystems is formulated in terms of state vectors defined on spacelike surfaces that evolve unitarily in between measurements and are reduced at measurements. Unitary evolution is represented by
|ψ(x,t)> = [e^(-iH/t)]|ψ(x)>---2.1
where ψ(x) represents state at time zero t=0; we could calculate the transition amplitude between coordinates q` at time t` =0 to coordinates q`` at time t``, that is an equivalent summary of unitary evolution, viz.
< q``t``|q`t` > = < q``|e^[(-iH/(t``-t`)]|q`>
(Coordinate indices, which may refer to either particles or fields, are often omitted to keep the notation compact). In a sum-over-histories formulation of quantum mechanics such amplitudes, are specified directly as path integrals
< q``|q` > = ∫(q`,q``) δe^[(iS(q(τ))/h] ;
Here, S[q(τ )] is the action functional corresponding to the Hamiltonian H and the sum is over paths q(t) that start at q` at time t` , end at q`` at time t``, and are single-valued functions of time. In an abbreviated notation, we may write
∫ δq` e^(-iS(q(τ))/h)|ψ> -- 2.4
for the state vector |ψ(t)i that is evolved by the propagator (2.3). More explicitly, (2.4) stands for the state vector |ψ(t)i whose representative wave function ψ(q, t) = < q|ψ(t)> is
ψ(x,t) = ∫ δq( ∫[q`,`] e^(-iS(q(τ))/h))ψ(x,t) ;
In this way, quantum dynamics corresponding to unitary evolution is cast into manifestly spacetime form involving spacetime histories directly. This is an important advantage in dealing with spacetime symmetries such as Lorentz invariance.
Unitary evolution, however, is not the only law by which the state vector evolves in quantum mechanics. In the usual discussion, at an ideal measurement that disturbs measured system as little as possible, the state vector is instantaneously “reduced” according to the “second law of evolution”
ψ(x,t) ---> [P(α)|&pshi;(x,t)>]/[||P(α)|ψ(x,t)>||]
where Pα is the projection operator on the subspace corresponding to the outcome of the measurement and k · k denotes the norm of a vector in Hilbert space. This “second law of evolution” may not be needed to calculate the transition probabilities in scattering experiments but it is essential for calculating the probabilities of the sequences of observations that define the histories of everyday life such as the orbit of the earth around the sun. It is every bit as essential for the prediction of realistic probabilities as is unitary evolution. Using the two laws of evolution, the joint probability for a sequence of measured outcomes α1, · · · , αn at times t1, · · · , tn may be calculated. It finds its most compact expression in the Heisenberg picture:
||Pn(α(n)(tn).Pn-1(α(n-1)(t[n-1]).....P1(α[1](t1)| ψ(t)>||^2 ---2.7
Here, {P k αk (tk)} is an exhaustive set of orthogonal Heisenberg picture projections representing the alternatives αk in the set k at time tk. For example, the set of alternatives might be an exhaustive set of alternative regions ∆k αk of the coordinates q at time tk.
The second law of evolution can be expressed simply in sum-over-histories form if attention is restricted to alternatives defined by sequences of configuration space regions {∆1 α1 }, {∆2 α2 }, · · · at times t1, · · · , tn [4, 5]. The joint probability that the system passes through the particular sequence of regions α = (α1, · · · , αn) is
||∫[c[α] δq exp[-iS(q[τ])/h]|ψ(x,t)>||^2 ---2.8
where the path integral is over the class of paths cα that pass through the region ∆1 α1 at time t1, ∆2 α2 at time t2, etc. Eq. (2.8) is a unified expression for the two laws of evolution in quantum mechanics Eq. (2.8) is equivalent to (2.7) for alternatives defined by exhaustive sets of exclusive regions of configuration space at definite moments of time. In this case the Schr¨odinger– Heisenberg formulation of quantum mechanics and the sum-over-histories formulation coincide. But they are not fully equivalent because the effect of projections onto ranges of momentum, for example, cannot be represented as restrictions on a configuration space path integral as in (2.8). Probabilities for momentum measurements can be predicted using path integrals, but only approximately, by modeling a time of flight determination of velocity in configuration space terms [21]. The sum-over-histories formulation of quantum mechanics therefore deals directly and exactly with a more restricted class of alternatives at sequences of moments of time than is available from the transformation theory of the Schr¨odinger–Heisenberg formulation

Path integrals and quantum gravity

Path integrals or sum over histoies also play a role in some of the candidate theories for a theory of quantum gravity. In string theory, for instance, the probability of interactions of given string can be calculated as a path integral - in addition to all possibilities a string can travel throughout ambient space, the sum is over all possibilities the string can deform and wiggle on the way.
Also, in the context of what is called quantum cosmology, some proposals for how the universe might have begun (including its "initial state") are formulated using path integrals, with or without the mythical factor i. In this scheme, the probability of the universe evolving into a certain state results from the sum over all possibilities for how this evolution might take place - literally a sum over all possible histories of the universe from its initial state to the present. However, the mathematical foundation for defining path integrals where the curved space-time of general relativity comes into play leads to some problems that, as of yet, have not been resolved wholly.


Summary and result of path integral

We need to evaluate a path function. Path function is defined using F(X) . X is a sample path of points X = x1.x2.x3.......xn

summary of path integral
Using a path measure μ(X) we can easily evaluate the path integral.
Notes and additional remarks

Complex numbers

Complex numbers are a wonder. To define complex number have to first define real number. Real numbers are the numbers that lie on the real number line. We know what real number is. It is measurable number. It can be used to measure something or some quantity. For example we can measure the circumference of a circle or a square by measuring the radius of the circle or length of the square respectively. We can use a ruler to do that. The rulers are the examples of measuring instruments, which have real numbers inscribed on them. Imaginary number does not exist in usual sense. You can say you have two apples or three apples but you can not say you have some imaginary quantity of apples. It can not be a measurable quantity. It is just a mathematical construct to simplify calculation appeared in equations. It is expressed in the form a+ib (where i=squareroot(-1)). So imaginary number can be defined as a couple of real number a and b. There is nothing special about complex number except that it involves imaginary unit i. Electrical engineering theories involve complex quantity for many purposes.

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It all adds up in path integral

This might sound odd, but it is precisely the view adopted by quantum theory. Think of a particle of light coming to our eye from a lamp. Common sense might suggest that it simply travels in a straightforward way from the bulb to the eye. But to make correct predictions about the particle's behaviour, quantum mechanics must take account of all other possible paths too, including ones in which, say, the photon bounces around the walls thousands of times before reaching us.
This summation of all paths, proposed in the 1960s by physicist Richard Feynman and others, is the only way to explain some of the weird properties of quantum particles, such as their apparent ability to be in two places at once. The key point is that not all paths contribute equally to the photon's behaviour: the straight-line trajectory dominates over the curvy ones. Hertog argues that the same must be true of the path through time that took the Universe into its current state. We must regard it as a sum over all possible histories.

Some useful mathematical definitions and notations used in path integral formulation

Correlation function
The term correlation is borrowed from statistics. When a large number of data is involved we can work out a relationship between these. An example will be like following.
correlation function
N is here is the number of pair of points x and y.
Correlation function can be defined using integration too.
correlation function


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