What is Imaginary Time? Hawking's peculiar concept of time at the beginning of the universe, and the imaginary time we live in every day.

"At the beginning of the universe, imaginary time flowed."

In "A Brief History of Time," many must have wracked their brains upon encountering this strange idea: imaginary time. What exactly is this imaginary time, which becomes negative when squared?

Imaginary time for calculations to avoid singularities

First, let's organize Hawking's imaginary time.

According to general relativity, at the beginning of the universe, there should have been a "singularity" where density and gravity become infinite. This is a point where physics breaks down. Hawking and Hartle's no-boundary hypothesis avoids this singularity by considering that in the very early universe, the time axis takes on a quasi-time character, thus depicting a smooth, unbounded origin of the universe.

What needs to be understood here is the position of imaginary time in physics.

It is introduced as a computational technique obtained by mathematically transforming real time. The procedure called Wick rotation is a prime example, and physicists have carefully reserved judgment on the question of whether "imaginary time truly exists." It is an immensely effective tool. However, this does not claim that time actually has two axes—this is the imaginary time from the physics side.

Why does "now" have depth?

However, there is an argument that requests imaginary time from a completely different place.

The starting point is not cosmology. It is the structure of the experience of "now."

On the one-dimensional flow of time `t`, which we use in our daily lives and progresses unidirectionally from past to future, "present" is represented as a point with no width.

However, the actual "now" is not like that. What is currently happening still overlaps with what has just passed, and what is to come already intrudes. The reason a melody is heard as a melody and the ringing of a bell is heard as a sound is because of this coexistence of phases.

And this coexistence does not emerge, no matter how finely we divide the t-axis. No matter how densely we arrange moments without width, thickness does not arise from them. A one-dimensional description of time is not closed to the "now."

Fifth paper by Muranushi: "Imaginary Time" ── T = t + it

Murakami's fifth paper, "Imaginary Time," begins from this situation.

A description that does not close requires only the minimal independent axis to complete it—the principle of minimal extension. This is an extension applied to time in the same way that a number system, which once did not close with only real numbers, closed as the complex plane by adding one orthogonal axis.

The consequence is that a complete description of time is given as the composition of the real time axis t and the imaginary time axis it.

T = t + it

The time we experience is always this two-dimensional structure T—complex time—and clock time t is merely its projection onto one dimension. The thickness of the "now" is first positioned on the it-axis.

The two imaginary times do not compete.

Thus, the two imaginary times are understood to be distinct.

Hawking's imaginary time is a computational technique in cosmology, introduced as a mathematical transformation of real time. The paper "Imaginary Time" states that it is an independent descriptive axis required by the structure of "now," and is not reducible to t. Their ranges, methods of introduction, and purposes are different. The two do not compete.

Why are the techniques of physics so effective?

Rather, the question is why the technique of imaginary time functions so effectively in physics. Is there something about the two-dimensionality of time structure itself behind its effectiveness—this question is left open in the paper.

Imaginary time placed at the beginning of the universe, and the imaginary time flowing in this very moment.

Imaginary time isn't a topic for distant cosmological discussions. It's about the thickness of "now," the very moment you are reading this sentence.

↓Yuma Murakami's Fifth Paper "Imaginary Time - Non-closure of Description and the Principle of Minimal Extension" is here↓

↓ For those who want to know more about imaginary time, check out the Theta Corridor now ↓

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