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

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

145 submissions , 122 unreviewed
3,930 questions , 1,398 unanswered
4,853 answers , 20,624 comments
1,470 users with positive rep
501 active unimported users
More ...

Path integral over topologies; occurence in literature / plausibility?

+ 2 like - 0 dislike
113 views

Suppose I have the vacuum state $|0>$ without loss of generality for quantum electrodynamics (with topological term proportional to $E B$) with the following additional feature:

The Hamiltonian $H(t) = H_{matter}(t) + \int_M d^3x (\frac{1}{2}(E^2+B^2) + \theta EB)$ is defined only within the region $M \subset \mathbb{R}^{3}$ and on the $N$ covering sets $G_i \subset \mathbb{R}^{3}-M,i \in \{1,\dots,N\}$ this operator does not exist. Therefore, $H$ must act on states $|0,G_1,\dots,G_N;t>$ which have these covering sets as an additional degree of freedom that form an orthogonal basis in Hilbert space; an inner product is one if the product is formed between two equal states and 0 otherwise.

Now one can try to construct the path integral; the derivation of the path integral for quantum electrodynamics is straightforward, but one encounters with inner products $<0,G_1',\dots,G_N';t+dt|0,G_1,\dots,G_N;t>$. The inner product should be defined such that the set of all space topologies is generalized to spacetime topologies. For simplicity one can say that this inner product is a constant $a$ (can be determined by normalization) if there exist morphisms $M(G_i,G_j')$ that lie on intersections $G_i \cap G_j'$ (because one can define a functor from intersections in category of sets to chart transition morphisms in category of manifolds) for all $i\in \{1,\dots,N\}$ while $G_j'$ is in 4-dimensional neighorhood of $G_i$ and zero otherwise. Finally, (normalization factor $a$ can be absorbed into the integral measure) one would obtain the generalized path integral:

$Z = \int \mathcal{D}[G_i^*(t_n)] \dots \int \mathcal{D}[G_i(t_1)] \int \mathcal{D}[A_\mu] \int \mathcal{D}[\psi] e^{iS_{QED}} \mathbb{1}_{Consistency}$

Here, $\mathbb{1}_{Consistency}$ is the indicator function that satisfies above consistency conditions.

Questions:

- Does a theory like this exist in research literature?

- Makes such a theory, a sum over topologies, sense?

- Can the "sum over topologies" also be defined on algebraic varieties or other topological spaces?

asked Jan 18 in Theoretical Physics by PatrickLinker (30 points) [ revision history ]
edited Jan 18 by PatrickLinker

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$\hbar$ysics$\varnothing$verflow
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).
To avoid this verification in future, please log in or register.




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

Your rights
...