Wick Rotation as a Cooling Process: A Novel Perspective on the Origin of Quantum Mechanics and the Arrow of Time

1. Introduction

The quantization of gravity is a notoriously difficult problem. In quantum gravity, there exists a famous equation—the Wheeler-DeWitt equation—but its form is rather unusual [1,2]:

$$H\psi =0,$$

(1)

where $H$ is the Hamiltonian of general relativity and $\psi$ is the wavefunction of the universe. In contrast to the Schrödinger equation $\mathrm{i}\mathrm{\hslash }\partial \psi /\partial t=H\psi$, the Wheeler-DeWitt equation contains no time derivative. In other words, after quantizing gravity, the time variable $t$ disappears. To explain the absence of a time variable in the equations of quantum gravity, Connes and Rovelli proposed the thermal time hypothesis [3,4,5,6]. Specifically, using the Tomita-Takesaki theorem, they showed that the non-commutativity of quantum algebras generates a real modular flow parameter and then postulated that this parameter is the physical time of the system. Simply put, time is not a fundamental quantity but an epiphenomenon accompanying the non-commutativity of quantum algebras [5,7]. Consequently, the disappearance of the time variable after quantizing the Einstein field equations does not mean that time is truly absent; rather, it is hidden in the non-commutative structure of the quantum algebra.

It has been widely noted that the modular flow parameter in the Tomita-Takesaki theorem can be analytically continued to the complex plane [7,8], leading to the Kubo-Martin-Schwinger (KMS) condition, which serves as a criterion for thermal equilibrium [8,9]. Motivated by this observation, we have further argued [10] that the time variable $t$ should be generalized to a complex number $z=t+i\tau$, which can be viewed as a complex extension of the thermal time hypothesis. This complex generalization may offer a potential interpretation of the physical meaning of the Wick rotation. On the one hand, from the KMS condition, the imaginary part $\tau$ of complex time satisfies $\tau \in \left[0,\beta \right]$, where $\beta =1/k_{B}T$. This implies that as the system approaches absolute zero $T=0$, the imaginary time is no longer compact and becomes an effective temporal dimension. On the other hand, according to the thermal time hypothesis [6], the real part $t$ of complex time should be proportional to $1/\beta$. This means that at absolute zero $T=0$, real time becomes frozen. Combining these two aspects, when the temperature $T$ tends to zero, the real part $t$ of the complex time $z$ is frozen while the imaginary part $\tau$ dominates the temporal dimension—mathematically, this is equivalent to performing a Wick rotation. In other words, the Wick rotation technique used in physics may describe a cooling process down to absolute zero.

The Wick rotation is a widely used mathematical technique in quantum physics [11]. For example, to address the singularity problem in quantum gravity and cosmology, Hawking had to introduce imaginary time $\tau$ [12], i.e., transforming Minkowski spacetime into Euclidean spacetime via the Wick rotation $t\to -i\tau$, thereby making the path integral convergent. However, Hawking regarded imaginary time as a computational tool rather than physical reality. Furthermore, in quantum critical phenomena, imaginary time is also treated as an effective dimension [13,14,15,16,17,18,19,20]. When the system approaches absolute zero, it tends to equilibrium, where thermal motion ceases and classical time loses its meaning. In contrast, imaginary time $\tau \in \left[0,\beta \right]$ becomes an effective temporal dimension precisely because $\beta =\infty$ at absolute zero. Nevertheless, in current studies of quantum critical phenomena, imaginary time is still viewed merely as a mathematical trick and not as genuine time [13,14,15].

Building on previous work [10,16,17,18,19,20], this paper attempts to put forward a more radical perspective: imaginary time is not just a mathematical tool; it emerges as a genuine physical dimension near the absolute zero. In other words, the imaginary time $\tau$ that emerges at absolute zero and the real time $t$ that we perceive daily at finite temperature are essentially different parts of a unified complex time $z=t+i\tau$, both being modular flow parameters accompanying the non-commutativity of quantum algebras [2]. As we will see later, once this complex-time picture is adopted, it may provide answers to two profound questions:

(a). Why do microscopic particles obey quantum mechanics?

(b). Why does the arrow of time exist in the macroscopic world?

To address these questions, we first introduce the Tomita-Takesaki theorem in operator algebras and the resulting notion of complex time. Throughout this paper, we use natural units $\mathrm{\hslash }=\mathrm{c}=k_B=1$.

2. Complex-Time Picture

The Tomita-Takesaki theorem in operator algebras provides a potential mathematical foundation for the emergence of complex time in physical systems. Consider a von Neumann algebra $\mathcal{N}$ acting on a Hilbert space $\mathcal{H}$, and a cyclic and separating state vector $\left|\psi ⟩\right.\in \mathcal{H}$. Define an operator $S$ satisfying [4,7,8]:

$$SA\left|\psi ⟩\right.=A^{†}\left|\psi ⟩\right.,$$

(2)

for all $A\in \mathcal{N}$.

The operator $S$ in equation (2) admits a polar decomposition $S=J{∆}^{1/2}$, where $J$ is antiunitary and $∆$ is a self-adjoint, positive-definite operator. Based on this decomposition, the Tomita-Takesaki theorem asserts [4,7,8] the existence of a one-parameter group of automorphisms ${\alpha }_t:\mathcal{N}\to \mathcal{N}$ such that for every $A\in \mathcal{N}$,

$${\alpha }_{t}A={∆}^{-it}A{∆}^{it},$$

(3)

where $t$ is a real parameter.

If we define the modular Hamiltonian $H_{\psi }=-\mathrm{l}\mathrm{n}∆$, then equation (3) can be rewritten in the form of time evolution in the Heisenberg picture:

$$A_H\left(t\right)=e^{it{H}_{\psi }}A{e}^{-it{H}_{\psi }}.$$

(4)

with $A_H\left(t\right)={\alpha }_{t}A$.

Equation (4) shows that as long as the state vector $\left|\psi ⟩\right.$ is cyclic and separating, the modular Hamiltonian $H_{\psi }$ automatically generates a time evolution $t$ of operators. Connes and Rovelli thereby proposed the thermal time hypothesis [4]: the modular parameter $t$ is identified with physical time. In other words, time is not fundamental but an emergent byproduct of the non-commutativity of quantum algebras.

In the thermal time hypothesis, the time $t$ is taken to be real. However, in previous work [10], we analytically continued the parameter $t$ to the complex plane $z=t+i\tau$ and regarded this complex parameter $z$ as the fundamental time. This complex extension is self-consistent and directly leads to the KMS condition, which serves as a criterion for thermal equilibrium [7,8]. To see this, we extend equation (4) to the complex domain $t\to z$:

$$A_H\left(z\right)=e^{izH}A{e}^{-izH}.$$

(5)

Now define the thermal correlation function:

$${〈A_H\left(t\right)B〉}_{\beta }=Z^{-1}\mathrm{t}\mathrm{r}\left[e^{-\beta H}{A}_H\left(t\right)B\right],$$

(6)

with the partition function

$$Z=\mathrm{t}\mathrm{r}\left[e^{-\beta H}\right].$$

(7)

Applying equation (5) to equation (6) yields the KMS condition [10]:

$${〈A_H\left(t\right)B〉}_{\beta }={〈B{A}_H\left(t+i\beta \right)〉}_{\beta }.$$

(8)

The KMS condition (8) implies that the imaginary part $\tau$ of the complex time $z=t+i\tau$ has the domain $\left[0,\beta \right]$. Furthermore, according to the thermal time hypothesis [6], $t\propto s/\beta$, where $s$ is the proper time. These two observations imply [10] that when the system approaches absolute zero, $\beta =\infty$, the real part $t$ of the complex time $z$ becomes frozen while its imaginary part $\tau$ extends into a full-fledged dimension. Mathematically, this is as if the complex time $z$ rotates from the real axis $t$ to the imaginary axis $i\tau$. Based on this observation, we propose a complex-time picture:

The Wick rotation from the real-time axis to the imaginary-time axis,$\mathit{t}\to \mathit{i}\mathit{\tau }$, describes a cooling process—from finite temperature down to absolute zero.

In simple terms, at finite temperature, spacetime is (3+1)-dimensional Minkowski, but when a physical system is cooled to absolute zero, it becomes a 4-dimensional Euclidean space, where imaginary time plays a role on an equal footing with space. Below, we will use this finding to explain the emergence of quantum mechanics and the problem of the arrow of time.

3. From Heat Diffusion to the Schrödinger Equation

Consider a system of particles at finite temperature ($T>0$), dominated by real time $t$, whose macroscopic dynamics obeys the heat diffusion equation:

$$\partial \rho /\partial t=D{\nabla }^2\rho ,$$

(9)

where $\rho$ is the particle number density and $D$ is the diffusion coefficient.

In this paper, we assume $\underset{T\to 0}{\lim }D\ne 0$. This is a central premise of our argument; if this premise does not hold, the subsequent reasoning becomes invalid.

Now we cool this particle system down to absolute zero $T=0$. According to the complex-time picture in Section 2, lowering the temperature to absolute zero requires the complex time $z$ to rotate from the real axis $t$ to the imaginary axis $i\tau$. Consequently, at absolute zero $T=0$, the heat diffusion equation (9) becomes:

$$i\partial {\rho }_c/\partial \tau =-D{\nabla }^2{\rho }_c.$$

(10)

Because of the appearance of the imaginary unit $i$ in equation (10), $\rho$ is extended to the complex domain, becoming a complex-valued function ${\rho }_c$. Its complex conjugate ${\rho }_c^{*}$​ satisfies:

$$i\partial {\rho }_c^{*}/\partial \tau =D{\nabla }^2{\rho }_c^{*}.$$

(11)

Let $\mathsf{\Phi }={\rho }_c^{*}{\rho }_c$. Using equations (10) and (11), one readily obtains a conservation equation:

$$\partial \mathsf{\Phi }/\partial \tau +\nabla ·J=0,$$

(12)

where

$$\mathit{J}=iD\left({\rho }_c^{*}\nabla {\rho }_c-{\rho }_c\nabla {\rho }_c^{*}\right).$$

(13)

Equation (12) yields the integral form:

$$\int \partial \mathsf{\Phi }/\partial \tau dV=-\oint \mathit{J}·d\mathit{S}.$$

(14)

Thus, $\mathsf{\Phi }$ can be naturally interpreted as a probability density, and equation (14) expresses probability conservation.

To see the internal consistency of the physical picture underlying the transition from equation (9) to equation (10) as the temperature is lowered to absolute zero, we first argue why equation (9) is inadequate at absolute zero. Equation (9) describes a diffusion process, in which the entropy of the system necessarily changes—it essentially describes an entropy-increasing process. However, according to the third law of thermodynamics, the entropy of a system at absolute zero is either zero or a constant, which contradicts the diffusion process described by equation (9). In contrast, ${\rho }_c$ in equation (10) is a complex function, so it does not describe a diffusion process but rather a wave-like process. Moreover, from equations (12) and (14), the time evolution in equation (10) ensures the conservation of the probability $\int \mathsf{\Phi }dV$. Therefore, equation (10) essentially describes some kind of change of state of the system, where the state is described by the complex function ${\rho }_c$, and the entropy of the system remains constant throughout the evolution.

Evidently, equation (10) is formally identical to the Schrödinger equation, which can now be interpreted as a diffusion equation in imaginary time. However, unlike the heat diffusion equation at finite temperature, the process described by equation (10) does not change the entropy of the physical system. To ensure constant entropy and probability conservation, the evolution of ${\rho }_c$ in equation (10) must be unitary. In other words, the emergence of a wavefunction or state vector in quantum mechanics is essentially required to maintain the physical validity of the diffusion process at absolute zero—preserving constant entropy so that the third law of thermodynamics holds.

Based on the above discussion, we infer that quantum mechanics is likely a phenomenon that manifests near absolute zero, where the imaginary time dimension emerges. This implies that quantum effects become more pronounced as absolute zero is approached.

Furthermore, since temperature is fundamentally a statistical quantity and is meaningless for an indivisible, isolated single particle (such as an electron or a photon), such single particles are expected to perfectly obey the imaginary-time diffusion equation, i.e., the Schrödinger equation. However, if such a particle couples to a thermal environment and thereby acquires a statistical temperature, the imaginary-time effect will be gradually suppressed or even dissolved — this can be viewed as a decoherence process.

4. Arrow of Time

In the preceding section, we have argued that the heat diffusion equation in real time at absolute zero would violate the third law of thermodynamics. However, if we adopt our proposed complex-time picture, when a physical system is cooled to absolute zero, time must rotate from the real axis to the imaginary axis. At that point, the heat diffusion equation naturally transforms into the Schrödinger equation, thereby rendering the third law of thermodynamics valid—ensuring constant entropy at absolute zero.

In this section, we further demonstrate that the arrow of time is essentially a manifestation of the real part of complex time, whereas the imaginary part of complex time carries no arrow of time. This may explain why the arrow of time exists in the macroscopic world but is absent in the microscopic world.

Applying the transformation $t\to -t$ to the heat diffusion equation (9) changes its form:

$$\partial \rho /\partial t=-D{\nabla }^2\rho .$$

(15)

This shows that the heat diffusion equation (9) is not time-reversal invariant. Indeed, because it describes the finite-temperature case where a thermal environment is present, the entropy of the system changes, and consequently an arrow of time emerges.

Now consider the absolute zero case, where the heat diffusion equation (9) becomes the imaginary-time Schrödinger equation (10). Performing time reversal $\tau \to -\tau$ on this equation, while simultaneously taking the complex conjugate of ${\rho }_c$, leaves the form of equation (10) invariant

$$i\partial {\rho }_c^{*}/\partial \tau =-D{\nabla }^2{\rho }_c^{*}.$$

(16)

This implies that the heat diffusion equation (9) in imaginary time, or equivalently the Schrödinger equation (10), is time-reversal invariant. That is, there is no arrow of time in this case. In fact, at absolute zero, the entropy of the system should not change, so no arrow of time should exist. This observation highlights the internal logical consistency of the complex-time picture.

The above discussion suggests that temperature can be viewed as a “regulator” of the time dimension: at finite temperature, a thermal environment exists, the system is described by real time, and an arrow of time is present. When the system approaches absolute zero, real time becomes frozen, imaginary time emerges, and the entropy of the system remains constant, so there is no arrow of time.

As mentioned earlier, temperature is a statistical quantity that acquires physical meaning only in the presence of a large number of particles. For an isolated single particle, speaking of temperature is meaningless. In other words, a “large number of particles” essentially constitutes a “macroscopic” system, whereas an indivisible isolated particle (such as an electron or a photon) is a “microscopic” system. The former possesses a temperature, while the latter does not. According to the complex-time picture, a system with temperature is described by real time, whereas a system without temperature is described by imaginary time. Hence, it is natural to understand why microscopic particles obey the Schrödinger equation. Nevertheless, when such a particle couples to an environment at finite temperature, it becomes part of a composite “system + environment”—the composite system possesses a temperature (the environment temperature) and therefore must gradually transition to a real-time description. This “switch of the time dimension” manifests physically as the loss of quantum coherence and the emergence of an arrow of time, i.e., decoherence. Thus, from the complex-time perspective, decoherence is not a mysterious measurement process but rather a natural transition of the time description from imaginary time to real time.

5. Testable Theoretical Predictions

Based on the above discussion of the physical meaning of the heat diffusion equation (9) at absolute zero, we have found that to avoid violating the third law of thermodynamics, the real time in this equation must be rotated to imaginary time. This perfectly aligns with the complex-time picture proposed in Section 2: when the temperature of a physical system is lowered to absolute zero, real time must be Wick-rotated to the imaginary time axis.

Although the complex-time picture exhibits remarkable logical consistency, it remains a theoretical speculation. This picture requires that the imaginary part of complex time possesses physical reality, i.e., it exhibits spacetime symmetry properties similar to those of real time. To test the physical reality of imaginary time, we have derived, under the assumption of “imaginary-time Lorentz symmetry”, a universal imaginary-time relativistic equation describing zero-temperature phase transitions [10,20]:

$$\mathcal{F}\left[\varphi \right]={\left|{\partial }_{\tau }\varphi \right|}^2+{\left|\nabla \varphi \right|}^2-{\displaystyle \frac{T_c^2}{c_F^2}}{\left|\varphi \right|}^2+{\displaystyle \frac{T_c^2}{2c_F^2{\rho }_0}}{\left|\varphi \right|}^4,$$

(17)

where $c_F$ represents a material-specific constant. By “imaginary-time Lorentz symmetry” we mean that after performing a Wick rotation $\tau \to it$, equation (17) recovers the real-time Lorentz symmetry.

To test the validity of equation (17), we substitute it into the partition function (7) to obtain a path integral formulation [10]:

$$Z=\int {\left[\mathcal{D}{\varphi }^{*}\right]}_{\mathsf{\Lambda }}\int {\left[\mathcal{D}\varphi \right]}_{\mathsf{\Lambda }}{e}^{-{\int }_0^{\infty }d\tau \int d^D\mathit{q}\mathcal{F}\left[\varphi \right]},$$

(18)

where $\mathcal{F}\left[\varphi \right]$ is given by equation (17), and $\mathsf{\Lambda }=2\pi /a$ represents the momentum cut-off (with $a$ denoting the lattice spacing).

Using equation (18), we have made the following theoretical prediction [10,20]: For two-dimensional superconducting films, the zero-temperature coherence length ${\xi }_0$ and ​$T_c$ exhibit a crossover between two scaling regimes:

$$\left\{\begin{array}{cc}{\xi }_0\propto T_c^{-1},& T_c\ge T_{*}\\ {\xi }_0\propto T_c^{-1.34},& T_c\le T_{*}\end{array}\right..$$

(19)

The anomalous scaling exponent $-1.34$ in equation (19) is the key to testing the validity of “imaginary-time Lorentz symmetry” [10,20] and thereby confirming the physical reality of imaginary time $\tau$.

Furthermore, the argument that the heat diffusion equation (9) transforms into the Schrödinger equation (10) when cooled from finite temperature to absolute zero may also be tested. Specifically, one could cool a particle system initially at finite temperature down to near absolute zero and observe whether the system transitions from having an arrow of time to having no arrow of time

6. Conclusion

In this paper, using the Tomita-Takesaki theorem in operator algebras, we have argued that time is a byproduct of the non-commutativity of quantum algebras—the modular flow—and that it can be a complex number $z=t+i\tau$. Combining this insight with the thermal time hypothesis, we have proposed a complex-time picture: when the temperature of a system is lowered to absolute zero, real time $t$ becomes frozen while imaginary time $\tau$ emerges. Mathematically, this is precisely a Wick rotation. In this sense, the Wick rotation from the real-time axis to the imaginary-time axis, $t\to -i\tau$, describes a cooling process—from finite temperature down to absolute zero. Employing this complex-time picture, we have found that at absolute zero, the real time in the heat diffusion equation must be Wick-rotated to imaginary time, thereby transforming the equation into the Schrödinger equation. This perfectly ensures that the third law of thermodynamics is not violated—if the real-time heat diffusion equation were to hold at absolute zero, entropy could no longer remain constant. In contrast, the imaginary-time heat diffusion equation describes the evolution of a wave function and can maintain constant entropy.

Using the above findings, we have provided a unified explanation for the origin of quantum mechanics and the emergence of the arrow of time in the macroscopic world. At finite temperature, a thermal environment exists, the system is described by real time, and an arrow of time is present. When the system approaches absolute zero, real time becomes frozen, imaginary time emerges, and the entropy of the system remains constant throughout its evolution, so there is no arrow of time. More importantly, temperature is a statistical quantity that acquires physical meaning only in the presence of a large number of particles. For an isolated single particle, speaking of temperature is meaningless. In other words, a “large number of particles” essentially constitutes a “macroscopic” system, whereas an indivisible isolated particle (such as an electron or a photon) is a “microscopic” system. The former possesses a temperature, while the latter does not. According to the complex-time picture, a system with temperature is described by real time, whereas a system without temperature is described by imaginary time. This explains why microscopic particles must be described by the Schrödinger equation.

Nevertheless, when such a particle couples to an environment at finite temperature, it becomes part of a composite “system + environment”—the composite system possesses a temperature (the environment temperature) and therefore must gradually transition to a real-time description. This “switch of the time dimension” manifests physically as the loss of quantum coherence and the emergence of an arrow of time, i.e., decoherence. Thus, from the complex-time perspective, decoherence is not a mysterious measurement process but rather a natural transition of the time description from imaginary time to real time.