In order to predict how the universe started off, one needs laws which hold at the beginning of time. If the classical theory of relativity is completely correct, the beginning of time is *mathematically* a point of infinite density and curvature of space-time -- all the known laws of physics would break down at such a point. It's clear that we must formulate new laws to hold at these apparent 'singularities', but it's extremely difficult to construct laws at such badly behaved points. We also have no real guide from observations as to what these laws might be. What these singularities really indicate is that the gravitational field was so strong in the early universe that quantum gravitational effects become important: classical theory is no longer an accurate description of our universe. A 'quantum theory of gravity' (basically a theory of everything) must be introduced to truly understand the early universe.

Sadly, we do not yet have this complete theory which unifies quantum mechanics and gravity. However, we are fairly certain of some features that such a unified theory should have. One is that it should incorporate Feynman's proposal to formulate quantum theory in terms of a sum over histories. In classical theory, a particle has a single history. Feynman's approach suggests that particles actually follow every possible path in space-time, and with each of these histories there is an associated amplitude (the size of the wave) and phase number. The probability that the particle passes through some particular point is found by adding up the waves associated with every possible history that passes through that point. When carrying out these sums, however, you run into some technical problems. The only way to bypass this is by adding up the waves for particle histories which aren't in the 'real' time that you and I experience, but take place in what is known as imaginary time.

While the concept of a number being imaginary may sound strange, it is in fact a well-defined mathematical concept. The square root of -1 is an imaginary number dubbed i. The square root of -4 is 2i, -9 is 3i, etc. If we picture real numbers as going left to right on a number line, imaginary numbers go down to up (i.e., they are in a sense at right angles to one another). To avoid the technical issues associated with Feynman's proposal, we must compute in imaginary time. This results in the distinction between space and time disappearing completely (a Euclidean space-time). So, as far as quantum mechanics is concerned, we may use imaginary time and Euclidean space-time as a mathematical device to uncover details about real space-time. A quick way to see how imaginary time makes physics easier is by playing around with the Minkowski interval, ds² = dx² + dy² + dz² - dt². If we make the substitution for imaginary time, t=it', the interval in Minkowski space-time becomes ds² = dx² + dy² + dz² + (dt')² -- a sort of 4D Pythagoras' theorem, allowing for more elegant calculations.

The second feature we know must be a part of any ultimate theory is Einstein's idea of gravitational fields being represented by curvatures in space-time. In classical relativity, there exist many possible curved space-times, each corresponding to a different initial state of the universe. If we know the universe's initial state, we would know its entire history. In quantum gravity, there are many different possible quantum states for the universe. Again, if we knew how the Euclidean curved space-times in the sum over histories behaved at early times, we would know the quantum state of the universe. 

According to classical relativity, our universe has either existed for an infinite time, or else it had a beginning at a singularity at some finite time in the past. The theory of quantum gravity gives way to a third possibility -- since we're using Euclidean space-time in which the temporal direction is indistinguishable from the spatial, it is possible for space-time to be finite in extent and yet have no singularities that formed a boundary or edge. Space-time would be like the surface of the Earth, only with two more dimensions. This theory opens up a new prospect of there being no boundary to space-time and thus no need to specify the behaviour at the boundary. Under this 'no boundary' proposal and using the Earth analogy, the distance to the North Pole can be thought of as imaginary time elapsed. As you move south, the circles of latitude get larger, representing the (growing) spatial size of the universe . The universe would reach a maximum size at the equator and contract with increasing imaginary time to a single point at the South Pole. Even though the universe would have no volume at the North and South Poles, these points would not be singularities, in the same way that the North and South Poles on Earth aren't singularities.

Those who know me personally may be aware that I've been fascinated by the concept of time being imaginary for quite a while now. Don't worry if the idea of it all makes absolutely no sense to you -- even as a physicist, it kinda baffles me. In my opinion [currently], the need to make such a substitution is just due to the fact that our current theories are limited. There may be a new/updated theory in the future which does not require us to introduce imaginary time for things to work smoothly. Nevertheless, it is important to understand that an eventual theory of everything may still require the use of tricks and/or approximations since the behaviour of our universe is clearly very complex and difficult to fully comprehend. Ideally this ultimate theory turns out to be simple and elegant, but the true goal of physics is to produce one which accurately describes the cosmos we inhabit.

published: 20/03/23 by kaan evcimen