□h=-16πT

Newtonian mechanics is set in flat space-time, curvature is a tensor that describes the degree to which parrallelism is lost. Are you thinking of radius of curvature?

In no particular order Black Books Never Mind The Buzzcocks Black Adder QI Have I Got News For You I'd put Python in there but I've not seen much of flying circus, just the films.

Are you reading this from his "Road to Reality" or the second volume of his and Rindler's book on spinors/twistors?

Quantum Field Theory in Curved Space-time Not a bad link, has a bit on the Hawking effect in there.

Quantum field theory is a relativistic theory of quantum mechanics whose most important tool is Lagrangian and Hamiltonian field theory, though one can drop the Hamiltonian formalism and take the Lagrangian as the more fundamental quantity (as is done in the path integral method). In QFT, matter, described by a field, arises as the quantised excitation of the ground state, the ground state being the vacuum for a free particle, and matter can be coupled to an interaction, which itself can then be quantised.

Shit, sorry about that.

Have a look at Feynman's path or functional integral formulation of quantum field theory. The path integral gives the amplitude for a particle, described by a quantised field operator, to move between two given locations. The amplitude consists of contributions from every possible path a particle can take in a time T, a linear superposition of contributions from each path weighted by a pure phase term which involves the action. The formula is [math]\langle x_b|e^{iHT}|x_a\rangle =\int {\cal D}xe^{i \int d^4x {\cal L}/\hbar}[/math] Where [math]e^{-iHT/\hbar}[/math] is a "time evolution" operator that takes the position [math]x_a[/math] into [math]x_b[/math] in a time T, the functional integral [math]\int {\cal D}x[/math] runs over each path and [math]{\cal L}[/math] is the Lagrangian density of the theory. The Lagrangian is a fundamental scalar quantity describing the whole of the theory. Have a look at Severian's post for an example of a Lagrangian (the free Dirac field Lagrangian). The usefulness of such a formulation is it offers a simpler mechanism for calculating propogation amplitudes and such like in perturbation theory. Certain paths will contribute more to the integral above than others, thus tending towards the classical path assumed by the particle, ie. that which satisfies the principle of least action.

Have a look at Feynman's path or functional integral formulation of quantum field theory. The path integral gives the amplitude for a particle, described by a quantised field operator, to move between two given locations. The amplitude consists of contributions from every possible path a particle can take in a time T, a linear superposition of contributions from each path weighted by a pure phase term which involves the action. The formula is [math]\langle x_b|e^{iHT}|x_a\rangle =\int {\cal D}xe^{i \int d^4x {\cal L}/\hbar}[/math] Where [math]e^{-iHT/\hbar}[/math] is a "time evolution" operator that takes the position [math]x_a[/math] into [math]x_b[/math] in a time T, the functional integral [math]\int {\cal D}x[/math] runs over each path and [math]{\cal L}[/math] is the Lagrangian density of the theory. The Lagrangian is a fundamental scalar quantity describing the whole of the theory. Have a look at Severian's post for an example of a Lagrangian (the free Dirac field Lagrangian). The usefulness of such a formulation is it offers a simpler mechanism for calculating propogation amplitudes and such like in perturbation theory. Certain paths will contribute more to the integral above than others, thus tending towards the classical path assumed by the particle, ie. that which satisfies the principle of least action.

Have a look at Feynman's path or functional integral formulation of quantum field theory. The path integral gives the amplitude for a particle, described by a quantised field operator, to move between two given locations. The amplitude consists of contributions from every possible path a particle can take in a time T, a linear superposition of contributions from each path weighted by a pure phase term which involves the action. The formula is [math]\langle x_b|e^{iHT}|x_a\rangle =\int {\cal D}xe^{i \int d^4x {\cal L}/\hbar}[/math] Where [math]e^{-iHT/\hbar}[/math] is a "time evolution" operator that takes the position [math]x_a[/math] into [math]x_b[/math] in a time T, the functional integral [math]\int {\cal D}x[/math] runs over each path and [math]{\cal L}[/math] is the Lagrangian density of the theory. The Lagrangian is a fundamental scalar quantity describing the whole of the theory. Have a look at Severian's post for an example of a Lagrangian (the free Dirac field Lagrangian). The usefulness of such a formulation is it offers a simpler mechanism for calculating propogation amplitudes and such like in perturbation theory. Certain paths will contribute more to the integral above than others, thus tending towards the classical path assumed by the particle, ie. that which satisfies the principle of least action.

Given that a singularity is a point, its dimension would be zero. Sorted. And yeah, any theory with more than 4 dimensions is conjecture and to think otherwise is just retarded. These 4+ dimensional theories are too over popularised.

The formulation of quantum mechanics that I've read deals with commutators of generators of the symmmetries of the galilean group, and the physical interpretation of these generators. My education in QM isn't particularly great, and I can't be bothered to sit and work it out myself, but I was wondering what result is obtained if instead of the galilean group one uses the Lorentz group? Does this lead to anything familiar to QED or does it simply produce results not too different to those of non-relativitistic quantum mechanics? The derivation of QED I'm familiar with is through quantising Maxwell's equations of classical electrodynamics. Thanks guys

You need to solve Einstein's field equations for that one.

No, not really.

If that equation were true then the length [math]L_0[/math] would dilate, not contract, with increase in velocity. It's the forumula for time dilation that involves multiplication by [math]\gamma[/math], not that for length contraction ([math]L=\frac{L_0}{\gamma}[/math]). Length contraction from Wolfram

No, the light would not pass through a flame quicker. The speed of light is always constant.

Gravity warps space-time=>different metric to that of flat-space=>different length to that of flat-space.

That's why I gave an example of the book's rigor.

I haven't read Cartan's book on spinors, but I have read about a third of the first of Penrose's volumes and the exposition is very detailed and rigorous. For example, Penrose discusses every necessary and sufficient topological condition for a manifold to have a spinor structure as well as discussing any properties that do not allow a spinor structure, in excellent detail. Cartan's book is 176 pages, whereas the first of Penrose's volumes is about 400, so you could judge it just by size. I'm currently on chapter 3.2 (of 5) and he is only really just getting into the algebra of spinors. The previous chapters, which are pretty long, develop the geometry and topology and introduce methods from classical tensor analysis that can be applied to spinor analysis (such as contraction, and the invention of a new labelling notation called the "Abstract index formalism").

[math]e^{i\pi}+1=0[/math] or [math]R_{\alpha\beta}-\frac{1}{2}g_{\alpha\beta}R=8\pi T_{\alpha\beta}[/math]

In reference to the shapes in higher dimensions, they only involve spacial dimensions. Dimensions aren't necessarilly spacial.

Dimensions aren't just spacial. A dimension is an independant parameter, it could be momentum and velocity (as in phase space), the Euler angles representing orientation or anything that some function on a manifold may have as an independant variable.

You're welcome .

Does that answer your question, or would you like a better explanation?