Path Integrals
Posted by on Friday, May 11, 2018 Under: Particle Physics
In honour of the 100th anniversary of the birth of Richard Feynman, I will today present a very basic overview of one of the great ideas of modern physics which was developed and popularized by Feynman. And the first part of that theory is the path integral formulation of quantum mechanics.
By now quantum mechanics is firmly established as a confirmed and proven theory of nature. Even popular society has come to embrace some of the stranger aspects of quantum theory (though unfortunately they often get it wrong). It seems like a most bizarre theory, and the truth is that it is a bizarre theory. No one fully understands quantum mechanics.
However there is one beautiful interpretation of quantum mechanics which owes its existence to Richard Feynman. (In the interest of accuracy, I must add here that Feynman himself gave credit for this idea to Paul Dirac, who published it a decade earlier but whose work was not well known and never developed further). This is the idea of the path integral.
Some of you may remember your introductory calculus class and the nature of integrals. If you have some function of a variable, you essentially sum the value of the function at each value of the variable. This is used to calculate areas of shapes, volumes of objects, and distances traveled, among countless other myriad uses. Every serious mathematics student has spent countless hours calculating integrals.
Path integrals are one step more advanced. In ordinary calculus you start with a function of one or more variables (ie a real or complex number, or a vector formed of such numbers) that maps each value to a real or complex number, and then sum the value of that function over all values of the variable. In path integrals, the numbers are replaced with curves through spacetime or through some more general space, and the function is replaced with a functional that maps each curve (or path) to a real or complex number. The path integral is then calculated by summing the value of this functional over all possible paths that connect two given points in the space. This sum is called the path integral.
By now quantum mechanics is firmly established as a confirmed and proven theory of nature. Even popular society has come to embrace some of the stranger aspects of quantum theory (though unfortunately they often get it wrong). It seems like a most bizarre theory, and the truth is that it is a bizarre theory. No one fully understands quantum mechanics.
However there is one beautiful interpretation of quantum mechanics which owes its existence to Richard Feynman. (In the interest of accuracy, I must add here that Feynman himself gave credit for this idea to Paul Dirac, who published it a decade earlier but whose work was not well known and never developed further). This is the idea of the path integral.
Some of you may remember your introductory calculus class and the nature of integrals. If you have some function of a variable, you essentially sum the value of the function at each value of the variable. This is used to calculate areas of shapes, volumes of objects, and distances traveled, among countless other myriad uses. Every serious mathematics student has spent countless hours calculating integrals.
Path integrals are one step more advanced. In ordinary calculus you start with a function of one or more variables (ie a real or complex number, or a vector formed of such numbers) that maps each value to a real or complex number, and then sum the value of that function over all values of the variable. In path integrals, the numbers are replaced with curves through spacetime or through some more general space, and the function is replaced with a functional that maps each curve (or path) to a real or complex number. The path integral is then calculated by summing the value of this functional over all possible paths that connect two given points in the space. This sum is called the path integral.
While it may at first seem esoteric and academic, it has a very important application to modern physics. If you have a physical system in state A (which could include the position of a particle, or multiple particles, as well as the momenta, energy and other properties of the system), then the probability that it will evolve to a different state B is the square of the path integral over all paths or curves that connect the two states. The functional that is used is a bit more complicated, but for the advanced reader it is F(C) = eiS(C) where S(C) is the action of the system over path C connecting A to B. All of classical mechanics, electrodynamics, quantum mechanics and modern particle physics is derived from this simple law of physics. This is the heart of quantum mechanics.
Unfortunately there are also problems with path integrals. The most important is that, with a few specific exceptions, no one actually knows how to calculate a path integral. There are some very fundamental problems with calculating an integral over all paths that are not present in integrals over real numbers. In ordinary calculus we can define a tiny interval of numbers, dx, and make it as small as possible, but in path integrals we do not know how to define a small interval of paths. And even if we did, the functional F(C) for any realistic physical system is incredibly complicated and the integrals even more so. This is one of the great unsolved problems of modern mathematical physics, and is still waiting for some brilliant mind to solve it.
Fortunately there are still many calculations that can be performed using the path integral formalism without actually having to calculated individual integrals. Most particle physics calculations are performed as a series of approximations of the path integrals, and many of the most powerful theorems of quantum physics have been proved using the known properties of path integrals. As we shall see in the next article, the entire Feynman diagram method of studying particle reactions is an approximation of the path integral formalism.
Perhaps of greatest importance is the the path integral formalism gives us an intuitive understanding of quantum physics in particular, and of the laws of nature in general. It is a beautiful theory produced by a beautiful mind.
Unfortunately there are also problems with path integrals. The most important is that, with a few specific exceptions, no one actually knows how to calculate a path integral. There are some very fundamental problems with calculating an integral over all paths that are not present in integrals over real numbers. In ordinary calculus we can define a tiny interval of numbers, dx, and make it as small as possible, but in path integrals we do not know how to define a small interval of paths. And even if we did, the functional F(C) for any realistic physical system is incredibly complicated and the integrals even more so. This is one of the great unsolved problems of modern mathematical physics, and is still waiting for some brilliant mind to solve it.
Fortunately there are still many calculations that can be performed using the path integral formalism without actually having to calculated individual integrals. Most particle physics calculations are performed as a series of approximations of the path integrals, and many of the most powerful theorems of quantum physics have been proved using the known properties of path integrals. As we shall see in the next article, the entire Feynman diagram method of studying particle reactions is an approximation of the path integral formalism.
Perhaps of greatest importance is the the path integral formalism gives us an intuitive understanding of quantum physics in particular, and of the laws of nature in general. It is a beautiful theory produced by a beautiful mind.
In : Particle Physics
I am a theoretical physicist & mathematician, with training in electronics, programming, robotics, and a number of other related fields.
