It is there in the sum over paths. For every path there is an amplitude, and you sum up the amplitudes for all possible paths to get the net amplitude for whatever process you are looking at. This sum is the superposition.
Take the two slit example. Initial & final states |i> and |f>. If the slits are a & b, you can think of the sum over paths as implying the existence of an intermediate or middle state |m> = |a> + |b>, which is a superposition of |a> and |b>.
This last equation is in the spirit of a sum over paths. Here there are only two, but if you had three slits you would have three paths, and if you had two walls with 3 slits in the first wall and 4 slits in the second wall, there would be 12 paths.
Infinitely many walls with infinitely many slits in each –> Feynman sum over (horribly uncountable) infinitely many paths.
But infinitely many walls with infinitely many slits in each = no walls with any slits at all!
Of course you need a formula for the amplitude for a path to begin with. This is exp(iS{path]/hbar), as Prof. Ingold showed.
The math seems to be getting lost in the typing. Let me try pseudo-tex
Since the slits are apart,
\langle b | a \rangle =0.
Then,
Total amplitude for process
= \langle f | m \rangle \langle m | i \rangle
= \langle f | b \rangle \langle b | i \rangle + \langle f | a \rangle \langle a | i \rangle
= Amplitude for path thru slit a + ampl. for path thru slit b.
If you would interpret the propagator in terms of a probability for a particle to start in some point x_a and end in the point x_b, the probability of starting in some point and end in any possible point (by integration over x_b) should be one. This could be seen as if the propagator is normalized when you integrate over all possible endpoints:
| \int dx_b K(x_b,x_a;t) |^2 = 1
This is true for a free particle, but not for the harmonic oscillator. Should the propagator be normalized and if not, why is it then true for the free particle?
Could you describe how to extend the Metropolis algorithm for the Ising model to a general path integral? Which quantities should be compared at the end of every step? Which probability is used to accept or reject the update?
Could someone please add the reference that discusses the ‘strange’ mapping between propagators (between the free particle and harmonic oscillator, for example)? Does this mapping have a physical meaning at all or is it just mathematically possible?
Please have a look at PRA 94, 043628 (2016) and the path integral references cited therein. The propagator Mapping between the free particle and the harmonic oscillator is just a mathematical trick, but the paper cited above suggests that this might have interesting consequences for many-body physics.
Is it possible to deal with acceleration as well as velocity terms in the action (particularly in statistical field theory) and if so how? Is this still possible if you also include velocity
& acceleration dependent potentials?
for the bcs bec crossover: has Anderson theorem something to say about bec. In other words, what happens to effect of disorder as you go from bcs to bec?
In the saddle point approximation of the BCS theory the gap equation is depending on the chemical potential.
Why does the dependence of the order parameter on the chemical potential not enter in the calculation of the particle number?
Comment (23)
Matthias| August 26, 2019
Where do I see entanglement in the path integral approach?
James| August 26, 2019
Does the path integral require a first order differential equation in time?
Matthias| August 26, 2019
Where is the superposition principle in the path integral approach?
Nona Belle| August 26, 2019
It is there in the sum over paths. For every path there is an amplitude, and you sum up the amplitudes for all possible paths to get the net amplitude for whatever process you are looking at. This sum is the superposition.
Take the two slit example. Initial & final states |i> and |f>. If the slits are a & b, you can think of the sum over paths as implying the existence of an intermediate or middle state |m> = |a> + |b>, which is a superposition of |a> and |b>.
Since the slits are apart, =0. Then,
Total amplitude for process
=
= +
= Amplitude for path thru slit a + ampl. for path thru slit b.
This last equation is in the spirit of a sum over paths. Here there are only two, but if you had three slits you would have three paths, and if you had two walls with 3 slits in the first wall and 4 slits in the second wall, there would be 12 paths.
Infinitely many walls with infinitely many slits in each –> Feynman sum over (horribly uncountable) infinitely many paths.
But infinitely many walls with infinitely many slits in each = no walls with any slits at all!
Of course you need a formula for the amplitude for a path to begin with. This is exp(iS{path]/hbar), as Prof. Ingold showed.
Nona Belle| August 26, 2019
The math seems to be getting lost in the typing. Let me try pseudo-tex
Since the slits are apart,
\langle b | a \rangle =0.
Then,
Total amplitude for process
= \langle f | m \rangle \langle m | i \rangle
= \langle f | b \rangle \langle b | i \rangle + \langle f | a \rangle \langle a | i \rangle
= Amplitude for path thru slit a + ampl. for path thru slit b.
Thomas| August 26, 2019
Questions raised in lecture 1a: https://github.com/gertingold/feynman-intro
Philip| August 26, 2019
Can we continuously deform (trap) a 3+1D system, with no knots, into a 2+1D system, with knots?
Feynman| August 26, 2019
Is there a general rule for the Kauffman invariant if you mirror a knot configuration?
Feynman| August 26, 2019
If you would interpret the propagator in terms of a probability for a particle to start in some point x_a and end in the point x_b, the probability of starting in some point and end in any possible point (by integration over x_b) should be one. This could be seen as if the propagator is normalized when you integrate over all possible endpoints:
| \int dx_b K(x_b,x_a;t) |^2 = 1
This is true for a free particle, but not for the harmonic oscillator. Should the propagator be normalized and if not, why is it then true for the free particle?
Mani| August 26, 2019
The propagators that we have seen in Lecture are gaussian-type. How could one evaluate Feynman propagators for non-quadratic potentials?
Anonymous| August 27, 2019
Could you describe how to extend the Metropolis algorithm for the Ising model to a general path integral? Which quantities should be compared at the end of every step? Which probability is used to accept or reject the update?
John| August 27, 2019
Is it worthy to compute the autocorrelation time during the simulation and save data space?
Salvador| August 27, 2019
Could someone please add the reference that discusses the ‘strange’ mapping between propagators (between the free particle and harmonic oscillator, for example)? Does this mapping have a physical meaning at all or is it just mathematically possible?
Axel Pelster| August 28, 2019
Please have a look at PRA 94, 043628 (2016) and the path integral references cited therein. The propagator Mapping between the free particle and the harmonic oscillator is just a mathematical trick, but the paper cited above suggests that this might have interesting consequences for many-body physics.
John Smith| August 28, 2019
Does it make sense to perform saddle point approximations on the fermionic path integral as ‘steepest descent’ does not seem well-defined.
bebe| August 28, 2019
Does the quantum monte carlo dynamic have physical meaning or is it purely artificial. What are the requirements for one or the other?
John Smith| August 28, 2019
Does the need for regularization suggest that steps in the calculation leading to the divergent expression are ill-defined (i.e. wrong)?
Boris Johnson| August 29, 2019
Is it possible to deal with acceleration as well as velocity terms in the action (particularly in statistical field theory) and if so how? Is this still possible if you also include velocity
& acceleration dependent potentials?
Anonymous| August 29, 2019
How can we relate large N theory to known theories, when all known theories have small N = 0, 1, 2, 3, … ?
Jhonatan| August 29, 2019
for the bcs bec crossover: has Anderson theorem something to say about bec. In other words, what happens to effect of disorder as you go from bcs to bec?
Einstein| August 29, 2019
In the saddle point approximation of the BCS theory the gap equation is depending on the chemical potential.
Why does the dependence of the order parameter on the chemical potential not enter in the calculation of the particle number?
Jean Zinn-Justin| August 29, 2019
I suggest one simple question and two exercises for the participants:
Why is $r_c$ on page 15 negative?
Do the explicit calculations for $d=0$ and $d=1$. Discuss the results of $d=1$.
Mani| August 30, 2019
Why graphene needs to be studied via relativistic quantum mechanics?