/sci/ - Science & Math

A field is a map from a vector space to a set.

>>9100768
In physics fields often nap to... well... a field (LOL I love the english language), a vector space, or a Lie group. Fields nornally interact via a group action.

>>9100794
>>9100768
If we are talking about a field in the mathematical sense, then it is a set with 2 operations that works similarly to addition and multiplication. These are the requirements of a field.

In the sequel, a, b and c are arbitrary elements of F.

>Associativity of addition and multiplication: a + (b + c) = (a + b) + c and a · (b · c) = (a · b) · c.
>Commutativity of addition and multiplication: a + b = b + a and a · b = b · a.
>Additive and multiplicative identity: there exist two different elements 0 and 1 in F such that a + 0 = a and a · 1 = a.
>Additive inverses: for every a in F, there exists an element in F, denoted −a, called additive inverse of a, such that a + (−a) = 0.
>Multiplicative inverses: for every a ≠ 0 in F, there exists an element in F, denoted by a−1, 1/a, called the multiplicative inverse of a, such that a · a−1 = 1.
>Distributivity of multiplication over addition: a · (b + c) = (a · b) + (a · c) .

>>9100794
>>9100799
But I mean in Quantum Mechanics!

>>9100804
Goddamnit dude

>>9100804
There you map an operator in Hilbert space to every point in position or momentum space.

>>9100768
I've always thought of a field as the negative gradient of a potential.

>>9100768
>>9100804
Classically, fields are smooth maps between manifolds that has non-trivial intersection with the solutions of the E-L equation.
In quantum mechanics, fields are smooth Hermiitian sections of a symplectic space satisfying certain quantization conditions.
In QFT, fields are operator-valued distributions satisfying certain regularity conditions.

>>9101514
How's the research going? Also you were one of the people who wanted to know about the model theory and operator algebra conference, it was a lot of fun and while they didn't allow video recordings they did post their lecture slides for the talks and the notes for the intro to model theory
https://www.math.uh.edu/analysis/2017conference-schedule.html
https://www.math.uh.edu/analysis/2017conference/slides/Oman.pdf
https://www.math.uh.edu/analysis/2017conference/Notes-Lupini.pdf
I was gonna post this in /mg/ but it is turning into a shit show

>>9101548
Glad to see that you remember our exchange. Thanks for the notes.
>How's the research going?
I've been busy with moving and administrative things recently since I'm starting a PhD soon. Now that things are starting to wind down I'll have more time to look at the things I'm interested in again, though the progress will be be slow since I can't take out any library books for now.
>I was gonna post this in /mg/ but it is turning into a shit show
Lmao tell me about it.

>>9101565
No prob. Ya got prelims next semester or just going straight into advanced topics? Also you might as well try looking on lib gen, was able to find a lot of good stuff there. Hopefully after the summer is over most of the of the really annoying shitposters will leave or at the very least be less active.

>>9101575
>Ya got prelims next semester or just going straight into advanced topics?
Prelims (i.e. core courses) were already taken care of by my MSc. It's all research from this point on, and maybe an interesting course here and there.
>Hopefully after the summer is over most of the of the really annoying shitposters will leave or at the very least be less active.
And we both know that the cuckold mods wouldn't do jack shit about that.
I'll try to keep discussions afloat in the physics general (if I ever get the time to make one). Hopefully the retards steer clear of it.

>>9101597
I really don't know what the mods are doing, I actually considered applying a few months back but I realized there's no cleaning this shit show, might as well just stay for the handful of good threads. Must be fun to no longer have any classes to take and just get to focus on research

>>9101610
Here come to our discord:
https://discord.gg/pqRWcf
>Must be fun to no longer have any classes to take and just get to focus on research
Well that should be expected.