How do I evaluate a path Integral?

[Anonymous No.7241784]

Ok so I am studying QFT and am learning path integrals. I understand there significance and how they are derived but I have no fucking idea how to evaluate one. Is it even possible to directly evaluate one?

[Anonymous No.7241904]

>>7241784
>learning QTF before learning vector calculus

path integrals are one of easier topic in that class. you should read through vector calc and learn it well before posting on /sci/ asking for help and looking stupid.

[Anonymous No.7241908]

>>7241904
also now that's i've insulted you, here's an actual explanation. let me pull up my calc 3/4 textbook

[Anonymous No.7241916]

>>7241904
>insulting me without knowing what a path integral is
lol

[Anonymous No.7241921]

>>7241908
>calc 4
What country are you from?
And sorry OP I can't help you with this, my time with calc 3 is over and I'm happy doing Chem

[Anonymous No.7241931]

think of a path integral as the area underneath a path as you follow it through space. the are evaluated by dotting the composition of the scalar function they are traveling through with the velocity of the path. pic related. the general form of the path integral in 3d space.
>>7241916
says the asshole who is trying to learn QFT without vector calc. also i wasn't insulting you for not knowing, i was insulting you for trying to learn shit out of order. i'm happy to help you actually.
>>7241921
usa , quarter system breaks vector calc into 2 parts.

sorry for the shitting definition, this is my laptop's webcam

[Anonymous No.7241939]

>>7241931
I can't tell whether you think i'm actually talking about calc iii shit but this is what a is meant by a path integral in terms of QM and QFT.

http://hitoshi.berkeley.edu/221a/pathintegral.pdf

[Anonymous No.7241962]

>>7241939
these are essentially the same thing. that was my point.

[Anonymous No.7241968]

>>7241962
No, no they are not.

[Anonymous No.7241977]

>>7241968
is big D not the matrix derivative?

[Anonymous No.7241982]

>>7241977
No a path integral mathematically is this...

<span class="math">
\int {Dx = \int\limits_{ - \infty }^\infty {\prod\limits_{k = 1}^\infty {d{x_k}} } }
[/spoiler]

[Anonymous No.7241984]

>>7241968
>>7241982
yeah that's pretty much the same, a different computation but it looks like the same basic concept. that i is the imaginary number right? doesn't that integral in the exponential for out to be a vector multiplied by the D matrix?

[Anonymous No.7241985]

>>7241984
work out to be*

[Anonymous No.7241992]

>>7241984
No, the integral is over a function space.

[Anonymous No.7241993]

>>7241904
>>7241908
>>7241931
>>7241962
>>7241984
Jesus fuck, are you stupid? Stop fucking posting.

Op, if you are curious read this: http://scitation.aip.org/content/aip/journal/jmp/1/1/10.1063/1.1703636

[Anonymous No.7241999]

>>7241993
LOL >>7241984 BTFO

[Anonymous No.7242128]

>>7241784
>MIT OCW
>Multivariable Calculus
>Line Integrals

[Anonymous No.7242165]

>>7241904
>>7241908
>>7241931
>>7241962
>>7241984
>Graduate level quantum physics, manifolds, differential geometry? Pshhh, I've done calc III, kid

[Anonymous No.7242191]

>>7241993
Thank You
>>7242128
I hope you are joking at this point

[Anonymous No.7242300]

>>7241784
Shit, I wish I rememberwd more. I remember that I needed Green's functions.
My class used Sakurai, but I found that Shankar to be more approachable.
I'd honestly just use your google fu.
Do you have a particular sample problem we could look at?

[mothergoose !!DUwV6rmVW4k No.7242540]

bumpin with a meme

[Anonymous No.7242743]

>>7241784

numerically? lol

but i'm surprised more '/sci/entists' aren't able to help you.

[Anonymous No.7242861]

Not directly. You're either going to have to use the symmetries of the Lagrangian to simplify the functional measure or put the fields on a lattice and let the lattice spacing go to zero afterwards.

[Anonymous No.7242867]

u need 2 take a mathematical methods course, mary boas is a common book.

[Anonymous No.7242932]

>>7241784
Path integrals are Ill defined objects that are useful as generating functional, but not so much when it comes to actual calculations.
Having said that, there are a couple of specific lagrangians were said integral converges. To work out the value, you're going to need to write down the measure explicitly, (with the limit and everything). You'll see that in some cases you are left with a pretty complex integral, that nonetheless converges and gives the same thing canonical quantization does. Google it

[Anonymous No.7244644]

>>7242743
Numerically is no fun

[Anonymous No.7244655]

>>7242867
>Boas
I'm not OP, using Boas, it's kinda easy really my prof told me to switch to Riley or Arfken.

[Anonymous No.7244688]

>>7241784
There are many ways. In my research, I had to evaluate a supersymmetric path integral over a symmetric space, of a nonlocal quartic interaction. You basically need like 4 different methods spliced together to do it.

Basically, here's the main methods:
- expansion about classical solution+path slicing or fourier mode expansion
- greens functions
- functional determinant + 'zeta function regularization'
- generating functionals
- symmetric space arguments
- atiyah singer index theorem (this one is the 'big guns' - basically always applies once you put things into the right form)
for information, try Springer's "Handbook of Feynman Path Integrals" - it basically outlines every single known method for doing this shit besides the index theorem.

Let me know if you have questions. If there's one thing I can do well in life, its calculate these bastards.

[Anonymous No.7244694]

>>7242743
you typically don't evaluate them numerically as an integral. you first do an expansion about the vaccua, then you solve some PDE numerically do get the eigenvalues, then you insert these into a functional determinant.

alternatively, you use mathematica to solve some complicated commutator relations in the case of fermionic fields.