Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,082 questions , 2,232 unanswered
5,352 answers , 22,785 comments
1,470 users with positive rep
820 active unimported users
More ...

  Does graviton loops affect the seperately covariant conservation of energy momentum of two noninteracting sectors of matter

+ 3 like - 0 dislike
2595 views
Consider the action $$\int \sqrt{-g}\left[R[g]+\mathcal{L}_{m1}(g,\psi_1)+\mathcal{L}_{m2}(g,\psi_2)\right]$$ Classically we have $$\nabla^\mu T^1{}_{\mu\nu}=0,\,\,\,\,\nabla^\mu T^2{}_{\mu\nu}=0$$ seperately. Point is that at the level of classical action $\psi_1,\psi_2$ does NOT interact directly but they both interact with gravity, that is to say, there are no terms of the form $\mathcal{L}_{int}(g,\psi_1,\psi_2)$, which, if present, would enable the two sectors to covariantly exchange energy-momentum and reduce the relation above to the weaker form $$\nabla^\mu(T^1{}_{\mu\nu}+T^2{}_{\mu\nu})=0$$ My question is whether quantum corrections can have such effects as well? Because naively it seems quantum mechanically many possible direct interaction terms between $\psi_1$ and $\psi_2$, even if no such terms are present in the classical action. Thanks a lot!

An answer to a simpler problem might equally be of value to me: the reduced case of abelian gauge coupling, the action $S=\int -\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\Psi_1 \gamma^\mu{D_\mu}\Psi_1+\Psi_2\gamma^\mu{D_\mu}\Psi_2$, whether $J_1$ and $J_2$ retains individual conservation.
asked Dec 11, 2014 in Theoretical Physics by xavier (85 points) [ revision history ]
edited Dec 11, 2014 by dimension10
What do you mean by direct interaction? There's no $\psi_1-\psi_2$ vertex in your lagrangian. However one can typically have loop diagrams with only fermion external lines, is that what you are thinking?

Hi @xavier, the \slashed commands weren't working in your post, so I replaced them with \(\gamma^\mu D_\mu\), which is the same thing. Hope it's OK.

yes exactly, because classically we'll have separate conservation laws for the two flows which is derivable from the fact that there's no $\Psi_1 - \Psi_2$ vertex in the lagrangian. I'm wondering if this is still true quantumally, or if there's still two separate conservation laws at all. The reason being what you said.
and what is the order of the violation of separate conservation
@xavier, My impression is that they will still be conserved separately in the quantized case, unless there are anomalies. I will write an answer shortly.
In your simple example I can see the symmetries which guarantee the separate conservation (modulo anomalies). In fact it is more traditional to talk not about the separate "electric currents" of $\Psi_1$ and $\Psi_2$ but about the total electric current which comes with the global gauge transformations (and you must keep it anomaly free) and under which the fields have the same charge, and the flavor current under which the fields are charged oppositely. I cannot see however why the stress tensors would be conserved separately with dynamical gravity (and if you want graviton exchange, you need dynamical gravity) even classically. Am I missing something obvious here?
Indeed the cases seem very different, the QED case has two obvious global symmetries guaranteeing the conservation. whereas in the gravity case there's no global symmetry, I find that confusing too. However, the conservation of each sector is a consequence of matter equation of motion and has nothing to do with whether gravity is dynamical (in my opinion). The conservation also follows from seperate coordinate invariance of the two matter parts of the action, we can apply the following procedure separately to each sector, because it is merely a (infinitesimal) change of coordinate. ($X$ is arbitrary vector field which vanishes on $\partial D$) \begin{align} 0&=\int_D L_X\mathcal{L}\nonumber\\ &=\int_D (E_\psi L_X\psi + T^{ab}L_X g_{ab})Vol\nonumber\\ &=\int_D (2T^{ab} \nabla_a X_b)Vol\nonumber\\ &=\int_{\partial D}boundaryterms-2\int_D (\nabla_a T^{ab})X_b\cdot Vol \end{align} Please keep me up to date with your opinion

1 Answer

+ 2 like - 0 dislike

There are two layers underlying the issue: The first layer is, formally we can show every classical conservation law has a quantum counterpart through, say, canonical quantization(see Weinberg's QFT chap7). The second layer is, if we go beyond the formal level, it is really that we would want the conservation law to hold and strive to make it hold in the quantized case too, unless there are obstructions such as quantum anomalies

On the surface of the issue, if we canonically quantize a classical Lagrangian, all conservation laws should be preserved, and this is fairly simple to see: the field operator equations have exactly the same forms of their classical versions, hence if any classical quantity has a vanishing 4-divergence, so will its quantized version.

However, reading between OP's lines, I suspect OP is having the misconception that a QFT can be defined by "a Lagrangian + a quantization recipe(canonical, path integral etc.)" This is not entirely true, a regularization and renormalization scheme must also be specified to make the QFT unambiguous. During this procedure, many symmetries are broken in the intermediate stages, and it is not a priori clear that the symmetries can all be restored after the procedure is finished.

The common practice is, through carefully controlling the regularize-renormalize procedure, we define a QFT preserving as many symmetries and conservations as possible. It is not always possible to preserve all the classical symmetries at hands, in such cases we will have the quantum anomalies.

Hence my earlier comment under your question:

My impression is that they will still be conserved separately in the quantized case, unless there are anomalies.

answered Dec 12, 2014 by Jia Yiyang (2,640 points) [ revision history ]
edited Dec 12, 2014 by Jia Yiyang
Thank you for the answer and the points you made, it's been beneficial~ In that case is there any quick way to see if a symmetry can be preserved in the quantum theory?
In the abelian case the separate conservation seems to remain true, because all vertices respects the conservation, because photons don't carry charge. However the nonabelian case look more subtle.

 In that case is there any quick way to see if a symmetry can be preserved in the quantum theory?

@xavier, This is beyond my expertise, I suspect there has been notable progress in detecting anomalies mathematically. Maybe @UrsSchreiber can shed some light on the question?

Is the @ functioning?

@xavier yes, and it works everywhere and has some additional useful features compared to SE for example..

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
p$\hbar$ysic$\varnothing$Overflow
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...