Starting Toward a Unified Theory

(Firstly, why does this erase my entire post every time I click anywhere in this typing field? ARGG infinitely infuriating)I have been able to gather a fairly general understanding many of the most important aspects of quantum mechanics and general relativity, in order to get a general understanding of string theory and other possible ToE's, but the mathematics behind these theories eludes me as I am immediately inundated with so many terms and ideas that are all interrelated that I have no idea how to actually start understanding even the most basic equations presented.Firstly, the Hamiltonian. How can i be a measure of energy if you can add or see them change through time, or add a "small perturbing Hamiltonian to calculate a more complex system." That doesn't seem to make sense! And then we have the Lagrangian which I am just totally baffled by. Is it a way to look at the time dependence on a system? Is it a symmetry or a law imposed on the system? What is it, why is it important, and how do you use or calculate it?What about the notation that inundates every equation involving a Hamiltonian and all the various states and vectors, eigen or otherwise, that you exist? Glancing down any page about QFT, GR or ST on wikipedia is completely indecipherable. I have many other questions, but maybe if I could just start here I might be able to make my own progress elsewhere.
- david (age 26)
Edmonds, WA, USA

First, we've passed your concern about the text box problems on to our web expert. We hope it will be fixed soon. Unfortunately I'm as bad at that sort of programming as at the deep physics which you ask about. I'll just give a first stab at some of your starter questions.

For an isolated system, the expectation value of the Hamiltonian doesn't change with time. In fact, the set of eigenvectors and eigenvalues and the squared amplitude of the state component along each eigenvector don't change with time. Thus the mean energy, its standard deviation, and all higher statistical moments don't change in time.

Various other quantities (e.g. position) do change because the interference between state components with different energies changes with time. That is often obscured in first courses on quantum mechanics, which focus too much on pure eigenstates of the Hamiltonian. Those don't do anything as a function of time, except rotate in a non-physical complex plane.

The business of adding small perturbing Hamiltonians (within the isolated system) is just a mathematical method for approximating solutions to equations that are too hard if handled in their full complexity. It doesn't change the physics described above.

Considering a small interaction Hamiltonian of the system with the outside environment does allow actual fluctuations in energy and other conserved quantities, through exchange with that outside environment.

I think the Wikipedia introduction to Lagrangians is good: . The Lagrangian is, like the Hamiltonian, just what you surmised-  a way to look at the time dependence of the system. One key physical point that doesn't seem clear in the Wikipedia description is why objects should follow trajectories that minimize the time integral of the Lagrangian, i.e. minimize the classical action. That integral is proportional to the change in the phase of the quantum wave from start to end of the trajectory. Having it minimized (or maximized) means that the components of the wave on a range of nearby trajectories are adding in phase, giving constructive interference. So that's where the wave mostly shows up.

The Lagrangian is not itself a symmetry, but it can have symmetries. In that sense it's similar to Newton's dynamical laws. They have certain symmetries, such as remaining true if all the objects they describe are displaced and/or rotated by the same amounts. Those symmetries lead to corresponding conservation laws.

That didn't tell you much but maybe it'll help a little.

Mike W.

(published on 12/15/2014)