In this post, I’ve put some more technical details about what the concept of imaginary time means, to help clarify the previous post about the Hartle-Hawking No Boundary Proposal<\/a>.\u00a0 If you don’t want to have to understand equations, skip this.<\/p>\n First of all, a bit of remedial math.\u00a0 There are a lot of functions which (even if they teach them to you in school as being functions of real<\/em> numbers) actually make sense when extended to complex numbers of the form $$z = x + iy$$.\u00a0 I already had to say something about complex numbers earlier in this series<\/a>.\u00a0 If you know how to add, subtract, multiply, and divide complex numbers, you can pretty easily make sense out of polynomial fractions like $$f(z) = z^3 + z \/ (z^2 – 1)$$, but you can also make sense out of things like sines and cosines and exponentials.\u00a0 For example, if we take an exponential of an imaginary number we get $$!e^{iy} = \\cos(y) + i \\sin(y).$$This formula allows you to turn all sines and cosines into exponentials, enormously simplifying trigonometry<\/a> by making it so you don’t have to memorize a bunch of weird trig identities.\u00a0 So even though they call them complex <\/em>numbers, they actually make your life simpler<\/em>!<\/p>\n So when you see something in a scientific equation like $$e^{ix}$$, that looks like an exponential, but the power is imaginary, that’s really something that’s spinning around in the complex plane as you change $$x$$, without growing or shrinking in its absolute size.\u00a0 It is a general rule that things which oscillate <\/em>in the real direction correspond to things which exponentially grow and\/or shrink in the imaginary direction, and vice versa.<\/p>\n This process of extending functions to the complex plane is called analytic continuation<\/em>, and functions which can be so continued are called (wait for it!) analytic<\/em>.\u00a0 (Not all functions are analytic: those which suffer from abrupt changes, like the absolute value function $$|x|$$, are not.\u00a0 $$|x$$ changes unpredictably at $$x = 0$$; if someone told you what it looks like for $$x < 0$$, and you tried to extrapolate it to $$x > 0$$ you’d guess wrong.<\/p>\n Now it turns out that there is a close mathematical connection between quantum mechanics <\/em>and thermodynamics<\/em> (a.k.a. statistical mechanics).\u00a0 Quantum mechanics is all about how the phase of a wavefunction oscillates around as time passes.\u00a0 The rate at which the phase spins around is proportional to the energy $$H$$ of the state, as told to us by Schr\u00f6dinger’s equation:$$!H \\Psi = i \\hbar (d\/dt) \\Psi.$$If you solve this equation, you find that a state with definite energy $$H = E$$ spins around as time passes like $$\\Psi(t) = \\Psi(0) e^{iEt\/\\hbar}$$, where $$\\hbar$$ is Planck’s constant.<\/p>\n On the other hand, statistical mechanics is all about thermal equilibrium states, and the rule of thermal equilibrium is that the probability to be in a given state falls off exponentially<\/em> with the energy.\u00a0 The probability is proportional to $$p = e^{-E\/T}\/Z$$, where $$T$$ is the temperature, and $$Z$$ is an extra random thing called the “partition function” you throw in to normalize the probabilities so they add up to 1. It turns out that states like these maximize the entropy <\/em>given how much entropy they have.\u00a0 If you squint these two exponentials they start looking quite similar to each other, if only you can accept the mystical truth that inverse temperature <\/em>is like imaginary time<\/em>: $$!1\/2T = it,$$where the factor of 2 comes from the fact that the probability is the absolute value squared <\/em>of the wave function.<\/p>\n If you start with an initial condition where all states have equal probability, and “evolve'” for a finite quantity of “imaginary'” time, you end up with a thermal state ( after normalizing the total probabilities to be 1 at the end).\u00a0 Better still, if you start with (almost any) state and evolve for an infinite <\/em>amount of imaginary time, you end up with the “vacuum” state of lowest energy, all other states being exponentially damped by comparison to that one.<\/p>\n Well, this may seem like a bit of mumbo-jumbo, but with the help of that complex number math I mentioned above, you can actually put it on a fairly rigorous footing, for ordinary QM systems, and even for quantum field theories.\u00a0 So of course, Hartle and Hawking had to be more bold than that, and try to apply this idea in the context of quantum gravity.<\/em><\/p>\n In quantum gravity (to the extent that we understand it), the dynamics are not governed by an ordinary Hamiltonian.\u00a0 Instead they are governed by a Hamiltonian constraint<\/a>: $$!H \\Psi = 0,$$also known as the Wheeler-DeWitt equation.\u00a0 This equation seems to say that nothing changes with time<\/em>, but it really means that the choice of time slice is arbitrary and has no coordinate-invariant meaning.<\/p>\n Now the Hartle-Hawking prescription is really just a clever way to calculate one particular state which (at the level of formally manipulating equations<\/a> that we can’t really make sense of) solves the Wheeler-DeWitt equation.<\/p>\n It tells us the wavefunction of the universe, expressing the “quantum amplitude” for any possible metric of space at one time to exist.\u00a0 (The quantum amplitude is just a term for the complex number saying what the wavefunction is for a particular possibility to occur.\u00a0 Take the absolute value squared and you get the probability.) Since there are many ways to slice spacetime into moments of time, all of them have to exist side-by-side in this wavefunction, late moments in time no less than early ones.\u00a0 That’s what it means to solve the Wheeler-DeWitt equation!<\/p>\n It’s not the only solution to the Wheeler-DeWitt equation, but it’s an especially nice one.\u00a0 In some ways it is like a “vacuum” state of the theory, one especially nice state to which others may be compared.\u00a0 (In other ways, it’s more like a thermal state, due to the fact that there is only a finite amount of imaginary time evolution, before one reaches the end of imaginary time).<\/p>\n In order to calculate the Hartle-Hawking amplitude that a given geometry for 3 dimensional space (call it $$\\Sigma$$) will appear ex nihilo\u00a0<\/em>(as it were), all you have to do is this:<\/p>\n 1. Consider the space of all 4 dimensional curved spatial<\/em> geometries whose only boundary is $$\\Sigma$$, Good luck!<\/p>\n <\/p>\n","protected":false},"excerpt":{"rendered":" In this post, I’ve put some more technical details about what the concept of imaginary time means, to help clarify the previous post about the Hartle-Hawking No Boundary Proposal.\u00a0 If you don’t want to have to understand equations, skip this. … Continue reading
\n2. For each geometry, integrate the total value of the Ricci scalar<\/a> $$R$$ over the 4 dimensional geometry, call that the action<\/em> $$S$$, and assign to that geometry the value $$e^{-S}$$.
\n3. Figure out how to integrate $$e^{-S}$$ over the infinite dimensional space of all possible 4 dimensional geometries.\u00a0 This requires choosing a measure <\/a>on this space of possibilities, which is quite tricky for infinite dimensional spaces,
\n4. Cleverly dispose of several different kinds of infinities which pop up, and
\n5. Consider all possible choices of $$\\Sigma$$ and figure out how to normalize it so that the total probability adds to 1 (nobody knows how to do this properly either).<\/p>\n