Imaginary time is a mathematical concept used in physics that represents time multiplied by the imaginary unit i (the square root of -1)[1]. While it may sound like science fiction, it’s a legitimate scientific tool used in special relativity, quantum mechanics, and cosmology.
Stephen Hawking explained imaginary time using a spatial analogy: “One can think of ordinary, real, time as a horizontal line. On the left, one has the past, and on the right, the future. But there’s another kind of time in the vertical direction. This is called imaginary time, because it is not the kind of time we normally experience”[2].
Recent experimental work has given physical meaning to this abstract concept. In 2025, physicists Isabella Giovannelli and Steven Anlage demonstrated how imaginary time manifests in the real world by measuring frequency shifts in microwave pulses[3]. Their groundbreaking experiment showed that imaginary time delays correspond to measurable changes in wave frequencies, proving that these mathematical constructs have observable physical effects[4].
Key applications of imaginary time include:
- Quantum Mechanics
- Used to calculate quantum states and predict system behavior
- Helps solve complex quantum mechanical equations
- Essential for understanding particle behavior at microscopic scales
- Cosmology
- Helps remove singularities in models of the universe
- Used in Hawking’s “no boundary proposal” for the origin of the universe
- Allows scientists to model the Big Bang without mathematical breakdowns[1:1]
- Special Relativity
- Appears in calculations involving spacetime intervals
- Helps describe the relationship between space and time
- Used in the Minkowski spacetime model[1:2]
Hawking noted that imaginary time is not merely a mathematical trick: “Imaginary time may sound like science fiction… But nevertheless, it is a genuine scientific concept. In fact, imaginary time is really the real time, and what we call real time is just a figment of our imaginations”[2:1].
The mathematical representation of imaginary time (τ) is obtained from real time (t) through what’s called a Wick rotation: τ = it, where i is the imaginary unit[1:3]. This transformation helps physicists solve complex problems that would be difficult or impossible to address using only real time.
Imaginary time is the time I get to spend playing video games after work and chores.
Physicists are either incredibly brilliant or completely insane. Or both. None of this makes any sense to me.
For people confused by the mathematics of imaginary numbers, the symbol i is just a shorthand rotational operator.
If something changes and comes back to it’s original state as if it rotated and you want to represent it in an equation, you use i.
Calling i imaginary makes it more mysterious than it actually is. The real world is filled with rotation.
Like particle spin?
Yeah. Although at the quantum level spin isn’t something physical like a ball spinning. It’s an intrinsic property. That is an electron changes its properties over time exactly as if it was spinning even though it’s not physically spinning as we understand it for large objects.
Ah I see. Thanks!
In fact, imaginary time is really the real time, and what we call real time is just a figment of our imaginations
Was almost falling asleep, damn
Oh that Stephen, he was such a prankster.
Huh, I thought ‘that’s brilliant, why shouldn’t/ofc changes in imaginary time affect measurable things’ … a second later, nah, that’s silly talk, how tf do you measure that??
Something related Measuring the imaginary time dynamics of quantum materials:
Theoretical analysis typically involves imaginary-time correlation functions. Inferring real-time dynamical response functions from this information is notoriously difficult. However, as we articulate here, it is straightforward to compute imaginary-time correlators from the measured frequency dependence of (real-time) response functions. In addition to facilitating comparison between theory and experiment, the proposed approach can be useful in extracting certain aspects of the (long-time relaxational) dynamics from a complex data set. We illustrate this with an analysis of the nematic response inferred from Raman scattering spectroscopy on the iron-based superconductor Ba(Fe1−xCox)2As2, which includes a new method for identifying a putative quantum-critical contribution to that response.
represents time multiplied by the imaginary unit i
Ok, i don’t really get it, but i get it.
(the square root of -1)[^1].
… aaaand it’s gone.
I think it’s easier to conceptualize that i^2 = (-1). So, if we have something like sqrt(-9), we can only describe the root as 3i, and the square is (3^2 )*i^2 = 9(-1). Alternatively sqrt(-9)=sqrt(9)*sqrt(-1).
If you look up “complex plane” you will see that one axis is real and one axis is imaginary, and the axes are perpendicular. If we have a term 3+4i (complex form a+b*i), then starting from the origin we move 3 to the right and 4 up. We often use the complex plane to describe periodic systems, where one full circle around the origin represents one period of oscillation. This allows us to visualize the magnitude of a complex value (sqrt(a^2 + b^2 )) and the phase angle (arctan(b/a)) of the state in a periodic system.
It gets a lot harder to conceptualize imaginary values when we talk about time, but consider that gravitational waves, which influence time, are also periodic in nature so they can also be described in complex form, and the resulting effect on time can also be described in complex form.
This is a really rough explanation and is definitely not 100% correct, but I hope it gets you closer to understanding the gist of imaginary numbers and why they are relevant here.
I respect what you’ve written, and thank you.
It’s the math, not the imaginary numbers that freezes my brain. I see 2+2 and my mind says “Orange!”
