Tuesday, 29 March 2016

The Path Integral Interpretation of Quantum Mechanics

The Path-integral Interpretation of Quantum Mechanics is a rigorous formulation of quantum theory that uses a particles path across space-time instead of wave functions.

Imagining a particle at some initial position in the (x,y)-plane, and we want to know what path it will take to some final position. By the classical least action principle, the particle will take a path between the two positions that costs the least energy. But, if the particle is a quantum particle, it’s not really localized at a point. Instead, the particle doesn’t take one path from the initial position to the final position, it takes all possible paths in a probability distribution space, or configuration space.



The basic idea then is to construct matrix elements of the time-evolution operator by summing all possible paths between two points and weighting the paths by the classical principle of least action. Remarkably, this gives identical results to solving the Schrödinger wave equation, but is much more concise a model.

The Hamiltonian generates the time evolution of quantum states. Ifis the state of the system at time t, then


This equation is the Schrödinger wave equation

From the Schrödinger wave equation we of course get the time evolution operator to work with.

Given the state at some initial time (t = 0), we can solve it to obtain the state at any subsequent time. In particular, if H is independent of time, then



By the homomorphic property of the functional calculus, the operator


This is the time evolution operator, or propagator, of a closed quantum system.

Given the time evolution operator we can construct a matrix element which is a probability amplitude for the transition of a particle from state X0 to Xt




The quantum action, S, is represented here as being the phase of each path being determined by ∫ L dt, for that trajectory, where L is the Lagrangian. Lagrangian mechanics is used for discrete particles each with a finite number of degrees of freedom. Furthermore, the action, S, is a relativistically invariant operator. [Ref-(1)].


Quantum mechanics is a probabilistic theory, hence the principle of least action has to be written to make a probabilistic statement. Suppose we have a quantum particle with some wavelength or frequency (either will do—since they’re inverses of each other) which we just measured at some initial point. We want to know the probability of finding it at some final point.

Each path is weighted with a probability factor ψ, which is the wavefunction, that depends on the action S for that path as:



                                                             
Each particle path is weighted by the wavefunction, ψ, which is scaled by Planck's constant h, for amplitudes summed up over the configuration space.

The total quantum amplitude is a sum over all possible paths, and is understood in the sense of quantum probability. The wavefunction ψ for any given path gives the quantum probability that the system will actually follow that particular path. The total quantum amplitude represents a quantum state all possible propagation paths. Under observation of course the system actually follows a particular one of these paths. The likelihood that the system will actually follow a particular path is determined by the wavefunction ψ for that particular path, which itself really depends on the fundamental action, S, associated for that path.


Each path the particle takes is equivalent to a probability wave as described by the  Schrödinger wave equation. Waves are in fact represented here in the complex plane as being circles traced out by the wave-form.

This phenomenon is inherently related to the nature of the complex plane, where the complex number z=exp(iθ)=cos(θ)+isin(θ) acts like a unit vector with a phase angle θ in the complex plane:
Euler's Equation can then be used to establish the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number x:


Where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively, with the argument x given in radians. 

The path of least action arises since the unit vector, z, tend to cancel out when the phase angles are out of phase with each other, and the vectors tend to add together when the phase angles are in phase with each other:

By taking each unit vector along probability wave represented by the path the particle takes, we form a series of arrows and make them oscillate around the complex unit circle at a frequency based on how difficult it is for a particle to travel through a given point—i.e. the harder the path to travel, the slower the arrow spins. (oscillating with frequency equal to the action.)

As discussed before that the particle takes all paths between the initial state and the final state. So we make our arrows follow each path and rotate them as we go along, as shown below.




Hence, in the path integral interpretation, the clockwork of unit vector arrows are along every possible path along the initial and final state (in this case 3 possible paths, red, green and blue). The paths are different only insofar as the arrows are rotated at different frequencies equal to the deviation from the path of least action, S. 

In other words the arrows are rotated with a frequency equal to that of the action imparted to deviate the particle from that of the least action possible. Therefore the path of least action itself should have the lowest frequency of unit vector rotation, where an increase in frequency corresponds to an increase in action.


In the path-integral view then, when the probability waves of particles transmitted in time and space are out of phase with each other they tend to cancel out (a). When the waves are in phase they tend to add together (b):



In the case of a single photon in the famous double slit experiment [Ref-(2)], which is the archetypal abstract model to test a theory in quantum physics, the photon acts like a particle that follows some actual path. The most likely path is the path of least action, but when we superimpose the behavior of many different individual photons we discover the interference pattern:




In the sense of quantum theory, every event is a decision point where the quantum state of potentiality branches into all possible paths.

Remember that since each particle path is weighted by the probability factor ψ, which is scaled by Planck's constant h, for amplitudes summed up over configuration spaces where Planck's constant becomes negligible the deviation from the least action (dS=0, i.e. a straight line) will become equally negligible. Hence we do not see quantum behavior in macroscopic configuration spaces, which is in everyday, ordinary, incoherent matter. 

Using path-integrals has revolutionized quantum dynamics by by-passing the task to solve the Schrödinger equation itself (which is possible only for few-atom systems) and instead giving a more mechanistic model of physics effects which are empirically non-intuitive, such as in the case of the double-slit experiment and its extensions. .

References:


[1] - Schwinger, J. ( 1951) Phys. Rev. 82, 914–927

[2] - QED: The Strange Theory of Light and Matter, Chapter 2.