2.) Feynman integral divergence有兩種，IR同UV，兩樣野好唔同。IR divergences通常因為低能量既時候wavefunctions可以form bound states、或者有soft-colinear emission，比較難搞。
3.) UV divergences因為極高能量既時候而家個theory唔make sense，應該有其他renormalizable theory (例如string theory 將個integration phase space cut剩個fundamental domain所以啲amplitude係finite)，而宜家個theory只係個UV theory既low energy effective theory。個low energy Lagrangian既divergent operators個coefficients應該用effective field theory既角度睇，詳情可以睇Wilson's Effective Action。
大致上係︰啲infinite coefficients唔係無限，只係啲operators with finite coefficients係零。
:^(
(好似係)
:^(
冷雨寒冬
2022-1-8 12:48:18
數學上嚟講 SO(3) 係一個 group 而L^2(R^3)係一個vector space
當你有一個group 同一個 vector space 你就可以討論 representation of a group in a vector space
Representation 係講緊對於每一個R in the group (而家嘅情況中個group 係SO(3)) 你都會有一個linear operator
P(R): L^2(R^3) -> L^2(R^3)
使得:
如果你有R, R' in SO(3) 而你想計下作用P(R')再作用P(R)會得出咩, 咁你可以先將R 同R' 乘埋(佢地可以相乘因為SO(3) 係一個group)再計P(RR')
上面用數學表達嘅話即係:
P(R) P(R') = P(RR')
當個vector space 有埋hermitian form, 咁unitary 係指啲linear operators P(R) 會preserve 埋呢個Hermitian form
Grothendieck
2022-2-2 19:34:59
I'm a mathematician, not a physicist, so here is a purely algebraic perspective: recall that a tangent vector is just an element v_p in T_p(M) and a cotangent vector is just an element p_v of the cotangent space T_p(M)*, which is by definition, just the set of homomorphisms from T_p(M) to R, i.e.e T_p(M)* = Hom(T_p(M), R), now since the momentum p is a homomorphism which takes a tangent vector and output a scalar, so it's a covector. But of course this doesn't give you any physical intuition of why momentum should be a cotangent vector.