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I'm trying to learn connection on principal fibre bundle. As far as I can see, the connection is just a given prescription for the displaced field/function on the base space to remains on the horizontal subset of the global space. [The global space and the base space are related by gauge symmetry equivalence class, the base space identifying two points related by a gauge transformation from the global space] In other words, a connection is an other word for parallel displacement. So it relates the field/function at position $x$ to the nearby one $x+\epsilon$ infinitesimal close to the previous position.

An interesting basic relation regarding connection is the covariant derivative, defined in Peskin and Schroeder (eq.15.4) [see the reference at the end] as

$$n^{\mu}D_{\mu}\Psi=\underset{\epsilon\rightarrow0}{\lim}\dfrac{1}{\epsilon}\left[\Psi\left(x+\epsilon n\right)-U\left(x+\epsilon n,x\right)\Psi\left(x\right)\right]$$

where $D_{\mu}$ is the covariant derivative, $\Psi(x)$ the field, and $n$ the vector along which the field is displaced, and $U\left(y,x\right)$ describes the propagation of the field $\Psi$ along the trajectory $x\rightarrow y$. For two distant points, $U$ becomes the Wilson line (eq.15.53, after a path is chosen).

If all what I said is correct up to now (please correct me otherwise), here is my question:

I'm wondering about the Green function, which is also a kind of prescription for the displacement of a field along a trajectory. Does it have something to do with a (mathematically properly defined) connection on a fibre bundle ? I've the feeling it gives a peculiar path among all the allowed ones. Is this feeling justified ?

More details about how I came to this question: In the Peskin and Schroeder, I've been particularly interested in the definition of a gauge transformation applied to their $U$. It's defined like (eq.15.3)

$$U\left(y,x\right)\rightarrow V\left(y\right)U\left(y,x\right)V^{-1}\left(x\right)$$

the $\rightarrow$ symbol meaning gauge transformation. This transformation law is the same as for a Green function defined as

$$G\propto\left\langle \Psi\left(y\right)\Psi^{\dagger}\left(x\right)\right\rangle $$

so I was wondering about the infinitesimal value of $G\left(x\rightarrow y \right)$ which is not clear because of the discontinuity ()say I'm talking about fermion). But still appealing to me. Also, it is clear that having the same transformation law is not sufficient to make any proof.

  • M.E. Peskin and D.V. Schroeder An Introduction To Quantum Field Theory Frontiers in Physics (1995)
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I would hazard a guess: no. First, a Green function is a measure of correlations between the field at different points, not displacement from one point to another (the interpretation as a particle propagator comes after the LSZ & invoking free asymptotic states business). Second, a (2 point) Green function is bi-linear in the field, unlike a covariant derivative or connection which is linear. So I'm leaning against the idea, but I don't really know. It would have to be some weird kind of bundle I've never seen before (which would certainly not be a hard thing to conjure up, I bet). –  Michael Brown Jun 18 '13 at 13:35
    
"I'm wondering about the Green function, which is also a kind of prescription for the displacement of a field along a trajectory". Which trajectory ? In the field realm, a 2-point Green function goes like like : $$G(x,y) \sim ( \frac{\delta}{\delta J(x)} \frac{\delta}{\delta J(y)} W(J))_{J = 0}$$ If you make a Legendre transform : $W(J) = \Gamma(\Phi) + \int ~dx J(x)\Phi(x)$, you will have $$G(x,y) \sim \frac{\delta \phi(x)}{\delta J(y)}$$ –  Trimok Jun 18 '13 at 14:20
    
@MichaelBrown That's precisely because it's bilinear that I have bad time to understand if my question makes any sense... :-( But still the correlation interpretation sounds appealing: it connect two different points I believe. I would like to understand how ? –  FraSchelle Jun 18 '13 at 15:59
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At least in QFT, one can imagine an instanton to quantify tunneling between different topological vacua, so it has to do with topology (or non-perturbative treatment), but not with perturbative treatment. So I do not see any relation. –  Trimok Jun 18 '13 at 16:28
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Yes, Green Functions are associated with perturbative treatment, and propagators and Green Functions are roughly the same thing (the propagator is a 2-point Green function). When you say that Green functions connect 2 points in space-time, I think it is not correct, a better point of view is to see Green functions as $ correlations$ of some field between its values at 2 different points. Instantons are topoligical "beasts". There is a conceptual difference between geometry and topology. For instance, the torus has a flat geometry, but its topology is not the same than the plane. –  Trimok Jun 18 '13 at 18:07
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up vote 3 down vote accepted

The answer is positive if you allow infinite dimensional manifolds, i.e., the bundle will not be over space or space-time but rather over the quantum state space which is an infinite dimensional manifold except for discrete level systems.

Actually, this is the basic description of the geometric formulation of quantum mechanics. Please see: Geometric Quantum Mechanics by Brodly and Hughston.

The basic construction is as follows: One considers the Hilbert space of the quantum system. The distinct pure quantum states of the system can be identified with the projective space of this Hilbert space, (i.e., the space of equivalence classes of points lying on the same line through the origin). Of course, this is due to the fact that an overall phase is unimportant in quantum mechanics.

In the discrete level case, this space will just a complex projective space $\mathbb{C}P_N$. In the more general case, the quantum phase space is a projective Hilbert space $\cal{PH}$ . This space has very remarkable properties:

First, it is a symplectic manifold, thus defines a classical dynamical system.

Secondly, it is Kähler, thus we can use the powerful tools of complex geometry in its analysis.

Thirdly, it is quantizable, i.e., if we basically apply the Dirac rules of canonical quantization to the classical system mentioned above, we will obtain our original quantum system. The state vectors of the quantum system are sections of a line bundle over the projective Hilbert space. This line bundle possesses a connection $A$ which has the properties required in the question. The curvature of this connection is the famous Fubini-Study form

Now, it is clear that picking up the projective Hilbert space as the base manifold is not a drawback , because in principle, we can choose to work in the basis of position states usually used in the formulation of Green functions.

The evolution of the quantum system will be governed by an operator on the Hilbert space whose matrix elements are given by the system's Hamiltonian $H$.

The above connection appears in the path integral formulation of the evolution operator:

$$ K_H(x, y, t) = \langle x | e^{iHt}|y \rangle = \int_{\cal{PH}}\mathcal{D}z e^{i\int_{x, 0}^{y, t} A(z) dz -H(z)dt}$$

Here $x$ and $y$ in the evolution operator correspond to space points and in the path integrals correspond to the position states.

Of course, the Green function can be obtained from the evolution operator by:

$$ G_E(x,y) = \int_0^{\infty} dt K_{E-H}(x,y,t) dt$$

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Sorry for my late comment: Thanks a lot for this description. Would you say that my question was stupid: a propagator has more to do with (a kind of) the Wilson path-integral than with a connection per se? Thanks again. –  FraSchelle Dec 16 '13 at 21:20
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As no one has bothered to formulate an answer, I'll take a shot at it, but no guarantees about the accuracy:

As far as I can see, the connection is just a given prescription for the displaced field/function on the base space to remains on the horizontal subset of the global space.

Actually, the connection defines the horizontal subspaces.

There are different ways to characterize a connection, eg as a subvectorbundle of the tangent bundle complementary to the vertical bundle, as the associated projector (which can be realized as a Lie-algebra valued 1-form in case of principal bundles - the gauge field), as a differential operator or as a prescription for parallel transport.

In contrast, Green's functions are correlation functions of a particular field. The two-point Green's function is just the propagator - the fundamental solution of the equations of motion.

When you use the propagator to 'displace' your field, you solve the equations of motion. When you use the connection (ie parallel transport the values), you end up with a constant field, for an in general path-dependent notion of 'constant'.

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