Tim Henke

2/25 Classically, you might have wanted info about speed, but in QM this is included in spatial info. Actually, particles can never be in such a precise position, but it's not so relevant and dealing with that hurts the exposition. The following can be done for any real state.

3/25 *Maths sidenote* the phase space is L₂(ℝⁿ;ℂ) and |q⟩ is represented as δⁿ(q-x) (Dirac δ). This is outside the space (reflecting its non-physicality), but also a reasonable extension. Such functions have "equal" contributions from all momenta (Fourier transform).

4/25 I will try to present an almost entirely elementary, conceptual explanation here. For more mathematical detail, one can read [1] for the exact same argument with more mathematical symbols or [2] for a more elaborate discussion, but using Euclidean time instead of Lorentzian.

6/25 Coming back to our original question, we see that the notation was a little sloppy. Let's rephrase our question as: what is ⟨q₂|eⁱᴴᵀ|q₁⟩? That is, starting with a state |q₁⟩ and letting it evolve for time T, what is the |q₂⟩-component of the resulting state?

7/25 The problem is that ⟨q'|eⁱᴴᵀ|q⟩ is in general difficult to compute. For a free particle (constant potential V), it would evaluate simply to exp[-im(q' - q)²/2T+iV] where m is the mass and m(q' - q)²/2T² is the kinetic energy to travel classically from q to q' in time T.

8/25 More generally H consists of a kinetic part and a potential part V. The particle is "free" when the potential is constant. If it's not constant, the evolution is really hard to work out. So we need a trick. We'll approximate it with tiny steps on which V is roughly constant

11/25 BONUS TWEET: if you're terminally category-pilled you may notice a functorial flavour here. Topological Quantum Field Theory studies functors from the cobordism category to the category of finite-dimensional vector spaces preserving the monoidal structure (disjoint union).

15/25 The calculation thus becomes ⟨q₂|eⁱᴴᵀ|q₁⟩ = ∫...∫∫ ⟨q₂|eⁱᴴᵗ|xₖ⟩...⟨x₂|eⁱᴴᵗ|x₁⟩⟨x₁|eⁱᴴᵗ|q₁⟩ dxₖ...dx₂dx₁ We integrate over each potential intermediate position at times nt = nT/(k+1). Now we can finally implement the trick from tweet 8.

16/25 We want to approximate V as roughly constant Mathematically, if t is very small (and energy bounded) we do this with the Baker-Campbell-Hausdorff formula Physically, if t is really small, the particle "won't have time" to travel anywhere where V is significantly different

17/25 "But wait!" I hear you sputter. "I thought we were integrating over all possible positions? Where does time to get somewhere come into play?" Well, yes, but.. Crazy paths in the integral are suppressed. Any path going really fast or far out the way or in general deviating

18/25 strongly from what a classical path would do barely contributes. This is, essentially, because the kinetic energy of a short path is roughly quadratic in its endpoints (think ½mv²). If (q' - q)/t = Δq/t is large then ⟨q'|eⁱᴴᵗ|q⟩ is proportional to exp[iΔq²/t + O(tΔq³)]

20/25 So paths for which a piecewise constant potential is a bad approximation are heavily suppressed. As we increase m (decrease t), the errors grow smaller but more numerous. This works out because the term neglected in the BCH formula is O(t²) = O(k⁻²) and there are k errors

21/25 Therefore the limit is well defined (for well-behaved V) and given by ∫...∫∫ exp[-im(q₂ - xₖ)²/2t + itV(xₖ) - ... - im(x₂ - x₁)²/2t + V(x₁) - im(x₁ - q₁)²/2t + V(q₁)] dxₖ...dx₂dx₁ as k -> ∞, using tweet 15, approximated with piecewise constant V as in 7.

22/25 We define the path integral as this limit and write it ∫𝒟x exp[-i ∫ mx'(t)² - V(x(t)) dt] where x is understood to range over all once-differentiable paths and x' is its derivative. Note how the argument of the exponential is a Riemann sum and thus becomes an integral.

24/25 Our justification for throwing out anything not at least once-differentiable is the following: viewing our paths as limits of sequences of points with xₙ -> x(nT/(m+1)), non-differentiable paths arise as the limit of "crazy paths", which are suppressed as seen in 17-19.

25/25 This way of seeing the paths as a limit also justifies the integral and derivative in the exponential. This is the Feynman path integral. For now, however, the symbols ∫𝒟x only denote an integral in name. We will discuss in a later part if it is actually an integral.