A global version of Günther’s polysymplectic formalism using vertical projections

Obstacles to the quantization of general relativity using symplectic structures

“…Therefore, although all the analysis to come can be generalized to apply to an arbitrary differentiable manifold I will keep things as simple as possible by making explicit use of the various nice properties of R 4 , including its canonical coordinates, metric, and (flat) connection. This analysis will therefore in many ways more closely resemble the long-known polysymplectic theory of [4] rather than my recent analysis in [5].…”

“…This too is more challenging in the case that M is a more generic (and, in particular, non-flat) differentiable manifold. See [5] for more details. )…”

“…It is, unsurprisingly, a section of T P ⊗ T P ⊗ T * M . (Yet again, this map is easier to define in global coordinates than in the general case; see [5] for details. )…”

“…Since our goal is to quantize general relativity on the simplest realistic space-time manifold R 4 , much of the sophisticated (and challenging) analysis of [5] is unnecessary. Therefore, although all the analysis to come can be generalized to apply to an arbitrary differentiable manifold I will keep things as simple as possible by making explicit use of the various nice properties of R 4 , including its canonical coordinates, metric, and (flat) connection.…”

“…The exception is Christian Günther's work, the so-called polysymplectic approach. Recently, I proposed a novel modification to the polysymplectic approach that continues to build upon its simpler geometric foundation, but is truly global in character [5]. This approach utilizes symplectic structures much more similar to that of the standard geometric approach to covariant Hamiltonian particle theory [6].…”