The Arrow of Time in Quantum Theory

Abstract

In Classical Mechanics, time is reversible, i.e., it implies no particular choice: only the observer knows in which direction it flows. The present article re-examines whether this remains true in Quantum Mechanics. In the context of Atomic Physics, it is concluded that the existence of an arrow of time depends on the manner in which the radiation field is introduced, which must be non-perturbative.

1. Introduction

This paper explores how the arrow of time, distinguishing between past and future, appears in Quantum Mechanics. It is not present in all formulations of the theory. In particular, it is absent in the context of Schrödinger-type equations, which are reversible in time. We examine the minimum requirement for this arrow to appear in the context of Atomic Physics. It is shown to be related to the non-perturbative introduction of the radiation field, because of the specific form of Planck’s Black-body Radiation law. It is also possible to introduce a direction of time by postulate in Scattering Theory, which creates initial and final states by definition, but this provides no intrinsic reason for the existence of an arrow of time.

The nature of time in Elementary Quantum Mechanics remains somewhat mysterious because, in contrast with the other observables, it is in no way different from time in Classical Mechanics. It appears as a variable in the Schrödinger and Dirac equations but is not associated with a Hermitian operator and remains unquantized. It is also reversible, in the sense that stable eigenstates occur. This initial formulation of Quantum Mechanics is referred to below as “elementary” in the sense that it does not include the radiation field, which remains external to the Schrödinger equation.

Nonetheless, Quantum Mechanics is applied to situations in which the distinction between past and future is essential, for example, in the decay of excited atomic states. It is therefore important to explore how and when the arrow of time appears as an inherent property of the theory and what minimum requirement this implies in its fundamental principles.

In the present paper, we briefly revisit how the theory is set up in order to identify what forces the distinction between past and future in Atomic Physics.

2. Background

As is well-known, time is an unquantized variable in Quantum Mechanics. It flows continuously ‘like a river’ in the sense implied by Newton, and is treated like a classical variable in Quantum Mechanics [1,2], which is a unique feature. It appears through derivatives (i.e., infinitesimal intervals) in both the Schrödinger and the Dirac equations. One might be tempted to turn to the Theory of Relativity for further insight, because the Dirac Equation is obtained by imposing the Lorentz transformations on elementary Quantum Mechanics [3]. However, as Dirac himself observed, his equation is not truly relativistic, since it remains a single-particle equation. Furthermore, relativity is none too helpful in this respect because space and time are regarded as different dimensions of a single entity, space-time. Since space has no preferred direction, this provides no way of associating an arrow only with time.

More generally, there are few situations in science that impose an arrow to distinguish between past and future. One has the second law of thermodynamics (extending also to Information theory), through the persistent growth of entropy [4]. Another example is Hubble’s law in Astronomy [5], leading to the continuing expansion of the Universe after the ‘Big Bang’ [6]. Finally, one thinks of Darwin’s law of biological evolution, in his theory of species [7], a subject completely outside the scope of the present Comment.

Elementary Quantum Mechanics appeals to a number of fundamental principles: the quantisation of Energy (to introduce Planck Constant ℏ), the representation of physical ‘observables’ by Hermitian operators acting on wavefunctions whose modulus squared yields a probability of occurrence, the act of measurement, resulting in combinations of eigenvalues of these operators, the physical interpretation of the commutation properties of operators and the postulate of the Uncertainty Principle, all of which are given as fundamental axioms. The steps involved are fully described in refs [8,9]. The reader will note that the form of Planck’s Black Body radiation law (also regarded as fundamental to Quantum Mechanics because it introduces the constant ℏ) is not actually required at this point to set up the elementary form of the theory. This is because, strictly, the Schrödinger and Dirac equations apply only in the absence of the radiation field.

These axioms alone are still insufficient to set up elementary Quantum Mechanics. It is also necessary to find appropriate mathematical forms for each of the Hermitian operators representing a given physical variable, without which Quantum Mechanics would have no inherent structure. To assist in this process, Bohr and Sommerfeld proposed the so-called Correspondence Principle, based on the idea that, in the ‘classical limit’ (i.e., for ℏ → 0) results obtained by Quantum Mechanics should merge seamlessly into those obtained by Classical Mechanics for the ‘corresponding’ system. Specifically, they proposed their rule in reverse (i.e., going from classical to quantum physics by integration around a closed classical orbit, according to the formula:

$${\oint }_{C}p dq=\left(n+{\displaystyle \frac{1}{2}}\right)\mathit{\hslash }\omega$$

where p and q are conjugate variables in the Hamiltonian, n is referred to as the principal quantum number and $\hslash \omega$ is the quantum of energy. This approach works perfectly well for ideal systems such as a simple harmonic oscillator or a ‘free’ atom with a Newtonian central field, in which case the orbits in phase space close exactly. It no longer works for more complex systems, such as a ‘real’ Rydberg atom, whose field is not Newtonian, since the atom then radiates and has no stable orbits above the ground state. To create an ideal atom, one begins by turning off both magnetism and the radiation field. Both are inconvenient because they prevent orbits from closing. Hydrogen then becomes the microscopic analogue of Newton’s two-body planetary system, with a central and purely inverse square law of force. Exact solutions for this case enable the Correspondence Principle to apply, i.e., the quantum system exhibits specific closed orbits in the classical limit.

Problems appear, however, as soon as greater complexity is introduced. First, [10] the three-body problem of classical physics cannot be solved in closed form, because the orbits in this case never close. They are chaotic. This precludes exact solutions for classical few-particle systems with three or more constituents. Such systems are ‘non-integrable’ and must be handled perturbatively. A second important example of a classically chaotic system is the pendulum with a magnet (the quantum analogue of which is a Rydberg atom in an external magnetic field). In the classical limit, if the pendulum is supposed to ‘write’ on a piece of paper, however long one waits, the path followed by the pen would never repeat. Here again, the Correspondence Principle fails to take us fully from the classical to the quantum system. Somehow, time is involved in this failure: in a potentially chaotic situation, it might take an infinite time to decide whether an orbit is about to close or not. If we attempt to apply the principle in reverse for such cases (i.e., going from quantum to classical physics), it is no longer clear what the classical limit would be, i.e., whether the whole of classical physics can be recovered, or only the part involving orbits that do close. Again, ‘quantum chaos’ does not exist for another reason: the Uncertainty Principle does not allow one to verify orbital closure because of the granularity of phase-space. Other examples exist of the limitations of the Correspondence Principle: for example, there is no semiclassical limit for the Dirac equation, simply because there is no such thing as Spin in classical physics.

In the classical limit, a two-body planetary system exhibits stable closed orbits at any energy up to the escape threshold. Quantisation results in stable orbits of infinite duration only at specific energies given by the eigenstates. However, this infinite duration has the same origin as in Newtonian mechanics, implying that there is no arrow of time. The dynamics are fully reversible. When invoking ‘stable orbits’ for an atom in the classical limit, it is important not to forget that, in reality, there are no stable classical orbits at all in this problem unless one turns off the effects of the electromagnetic field. Thus, the Correspondence Principle is ultimately flawed (as first noted by Einstein). Despite this limitation, it yields a very powerful computational approximation for high quantum states of Rydberg electrons once chaotic orbits are excluded (see, e.g., [11]), but this method cannot provide any information on the arrow of time.

Finally, we come to the ‘Time-dependent’ version of the Schrödinger equation: This is not really a different equation, but rather a mathematical extension of the original Schrödinger equation, obtained by writing the energy explicitly as its operator, namely a derivative with respect to time. In terms of basic principles, it contains no more than the original equation and so contributes nothing further concerning the arrow of time. As will be seen in Section 2 and Section 3 below, it nonetheless provides the basis for perturbation theory leading to some further useful insights.

For the whole of this Section 2, which relates to the Schrödinger equation applied to bound states, one can equally well apply the general argument that the associated Lagrangian is fully time-reversible. Indeed, bound states last forever in this situation, with no beginning or end.

3. The Wigner Time Delay in Scattering Theory

The previous section on the Schrödinger equation relates to bound state solutions, which are stable in the absence of the radiation field and therefore have no beginning or end. The present section on scattering theory considers continuum solutions of Schrödinger-type equations for particles that can be separated, leading to the possibility of defining initial and final states and of computing ‘Wigner delays’ due to scattering.

Since Schrödinger-type equations have no preferred direction of time, it is not inconsistent to impose the existence of an arrow of time as an additional independent postulate. This is precisely what Wigner does [12] in Scattering Theory, by requiring that the scattering process should lead from an ‘initial’ to a ‘final’ state. In other words, this approach postulates an arrow of time, which, however, is not an intrinsic property of the system unless this assumption is made. The advantage of doing this is to bring situations in which particles are separated from each other in initial or in final states (or both) within a formalism based on Schrödinger-type equations, i.e., to provide a general and direct extension of the theory to the continuum states above the threshold in Atomic Physics by the introduction of the scattering matrices, which have many interesting and useful analytic properties in atomic theory [13], backed up by many observations.

Although the full nature of time still escapes analysis in this construction, an interesting new feature appears, known as the “Wigner time delay” [14], intrinsic to the scattering process. It appears as a result of the phase shifts induced during the scattering process itself. It owes its appearance to the fact that processes allowing detection of the scattering wave-packet require its phase to be stationary. Otherwise, the superposition of many different phases during scattering by a dispersive potential would blank out any signal by phase cancellation for both incoming and outgoing waves. Following the equations in [15,16], expressed in the notation of Landau and Lifshitz [8] with ħ = m = c = 1, the radial wave-packet is represented as

$$e^{i\left[kr-Et+\phi \left(k\right)\right]} \equiv e^{i\Phi \left(k\right) }$$

where k is the radial wave vector of the free electron, whose kinetic energy is k²/2 and $\Phi$ is defined as the total wave-packet phase, as distinct from φ the scattering phase, then the stationarity condition, with the Wigner time delay ${\tau }_w$ defined by ${\tau }_w$ = (1/k) ꝺφ/ꝺk (see [16] for details of the formalism) yields

$${\displaystyle \frac{ꝺ\Phi }{ꝺk}}=r-kt+{\displaystyle \frac{ꝺ\phi }{ꝺk}}=r-k\left( t-{\tau }_w \right)=0$$

In experiments on atomic photoionisation [15], the scattering phase is further augmented by the photoemission phase shift, given by the argument of the atomic dipole matrix element, resulting in a collective phase for the complete scattering process. To compute this atomic contribution, one appeals to perturbation theory within the framework of the time-dependent Schrödinger equation (Section 1 above) with a partitioned Hamiltonian of the form

$$i\mathit{\hslash } {\displaystyle \frac{ꝺ\mathsf{\Psi }}{ꝺt}}=\left\{ \mathrm{H}\mathrm{ₐ}\mathrm{ₜ}\mathrm{ₒ}\mathrm{ₘ}+\mathrm{H}\mathrm{ₚ}\mathrm{ₜ}\right\}\mathsf{\Psi }$$

where $\mathrm{H}\mathrm{ₐ}\mathrm{ₜ}\mathrm{ₒ}\mathrm{ₘ}$ stands for the field-free atomic contribution and $\mathrm{H}\mathrm{ₚ}\mathrm{ₜ}$ for the perturbative scattering term. There are many approaches within atomic theory to compute these terms, which, together with comparisons to observations in recent experiments, are fully discussed in the topical review [16] by Kheifets.

What has attracted great interest is the fact that extremely short time delays (in the attosecond range) can be involved. For example, Ne Schultze et al. [15] report a Wigner time delay of 21 ± 5 as (2.1 × 10⁻¹⁷ s). For comparison, the atomic unit of time (the time taken for a Bohr electron of lowest orbit to travel a Bohr radius of an H atom) is 2.42 × 10⁻¹⁷ s.

In practice, we note that what is measured in such experiments is not truly a time, but rather a photoelectron spectral phase and its energy derivative, converted into the Wigner time delay through the conceptual framework of scattering theory.

Again, the ordering of time is imposed at the outset through the assumptions of the Wigner formalism, and so the arrow of time is imposed by postulate rather than emerging as a consequence of the theory. One can apply, as for Section 2 above, the general argument that the associated Lagrangian is time-reversible and so it follows that there is still no arrow of time involved, despite the emergence of what is termed a “time delay”.

4. Introducing the Radiation Field

The next step in the present context is the introduction of the radiation field. In classical physics, radiation prevents the occurrence of stable orbits. The electron emits Synchrotron Radiation, as proved by Schott [17]. The intensity distribution of Synchrotron radiation as a function of frequency is completely different from that of Planck’s Black-Body Radiation law. In the quantum theory of the atom, radiation is introduced by using the dipole approximation within a theoretical formalism due to Heitler [18]. Basically, the interaction with the oscillating electric component of electromagnetic radiation is treated as a perturbation of the free atom using the time-dependent Schrödinger equation given above and this results in a formula expressing the probability per unit time of both excitation from a state i to a state j of the atom by absorption of a photon and also the inverse process, from j to i, stimulating the emission of a photon, both of which occur simultaneously in the presence of the radiation field.

It was shown by Sommerfeld and Schur [19] that this description via the dipole approximation remains incomplete, because it takes no account of the radiation pressure, or momentum of the photon, whose absorption or emission leads to recoil of the atom. They showed that the minimum requirement to describe the dynamics is to also include the quadrupole terms in the perturbative expansion. However, even in this higher level of approximation, the theory remains within the framework of perturbative extensions to the Schrödinger equation, which, as argued above, provide no new insight concerning the arrow of time.

The next step was introduced by Einstein [20], who considered the problem from a completely new angle, and by Milne [21], who elaborated Einstein’s argument in the context of stellar atmospheres. For the first time, Einstein introduced the radiation field in a non-perturbative way, by simplifying the atomic physics to just two active states [1,2] and placing two-level atoms in a bath of radiation in the absence of collisions. He argued that the only simple way for isolated two-level atoms to come into equilibrium with the radiation field is to apply what is called the ‘Principle of Detailed Balance’, namely that pairs of states should all be individually in equilibrium with each other in the presence of the radiation field, which is only possible if

$$g_1 B_{12}=g_2 B_{21}$$

where g₁ and g₂ are the statistical weights of levels 1 and 2, and the coefficients B₁₂ and B₂₁ are the probabilities per unit time for absorption of radiation from level 1 to level 2 and the probability of stimulated emission from level 2 to level 1 obtained from the Heitler theory in the dipole approximation. Up to this point, everything remains reversible, including the Principle of Detailed Balance. However, as argued by Einstein, this theoretical structure is not sufficient to recover a most fundamental property of radiation associated with the quantum theory, namely, Planck’s Black Body Radiation law. In order to recover this law, Einstein was obliged to introduce into the theory a new coefficient, which he called A₂₁, the probability per unit time of spontaneous decay from level 2 to level 1 in the absence of any external radiation field. He showed that, if

$$A_{21}\equiv {\displaystyle \frac{2h{\nu }^3}{c^2}} B_{21}$$

then Planck’s Black Body Radiation Law is recovered. The two rules connecting the A and B coefficients are referred to jointly as the Einstein–Milne relations. It is important to note that Einstein’s hypothesis was the first actual use of Planck’s Black-Body Law in setting up Quantum Mechanics, and was not needed for the Schrödinger equation, although this law is always cited as one of the fundamental principles of quantum theory.

The full significance of Einstein’s A coefficient was unknown originally, even to Einstein. Only later was it discovered that the spontaneous emission coefficient is a consequence of quantum fluctuations in the radiation field, in other words, that quantum field theory is responsible for its existence [22].

By considering the equilibrium conditions and applying Detailed Balance to two- and three-level systems, one finds that a population inversion requires at least three levels, and therefore that laser action by stimulated emission of radiation can only be achieved at the expense of energy. Laser light depends on the B coefficient and creates electromagnetic order, whereas spontaneous emission, depending on the A coefficient, results in the emission of incoherent light (disorder). Consequently, the statement that order is achieved at the expense of energy conforms to the second law of thermodynamics, which is not surprising since the Einstein coefficients were imported into Quantum Mechanics via Planck’s Black Body Radiation law, which itself results from thermodynamics.

The second law of thermodynamics, as pointed out in the Introduction, is one of the few laws of Physics that imposes an arrow on the direction of time. In the present instance, this appears as follows: if an isolated atom has been raised to an excited state, then it will decay back to its ground state by fluorescence in a time called the natural lifetime of that state, given by 1/A. This decay defines the arrow of time, since there exists no balancing inverse process.

The general argument of a time-reversible Lagrangian mentioned in Section 2 and Section 3 above does not apply here, because such a Lagrangian does not extend to the second law of thermodynamics.

In practice, there is, of course, no such thing as a fully ‘isolated’ atom in nature. Were it to exist, the Gedanken experiment would consist in exciting an atom initially, then turning off all external perturbations (collisions as well as the external radiation field) and seeking the moment when all fluorescence stops. This moment comes at a later time, so the arrow of time would thus be confirmed experimentally. However, it would not be sufficient to observe fluorescence from a single excited state, because individual excited states can also be populated by cascades from levels of higher energy, which could, in principle, result in errors of interpretation. This problem is a well-known feature of Beam-Foil Spectroscopy [23] and complicates the determination of natural lifetimes. Also, time is a property of the Universe, and so the arrow of time is not determined by the observation of a single atom, but rather by observing the collective behaviour of many atoms, which is why the arrow of time does not stop with the disappearance of fluorescence coming from a single atom.

5. Conclusions

The present paper demonstrates that the minimum requirement for the existence of an arrow of time in Quantum Mechanics is the non-perturbative introduction of the radiation field via Planck’s Black Body Radiation law and Einstein’s A coefficient. This conclusion is reached within the context of Atomic Physics, which provides the simplest conceptual framework to set up Quantum Mechanics. There of course exist other phenomena, such as radioactivity in the context of nuclear forces, which also imply an arrow of time, but whose analysis is more complicated because of the nature of the fields of force.

It is interesting to observe that time cannot be reduced to the properties of the variable t in elementary quantum theory, since this theory captures only its cyclic or repetitive nature, as for the two-body planetary or Newtonian problem in the absence of external perturbations. Thus, one can use the ‘atomic pendulum’ for states of very narrow natural linewidth to measure time intervals. However, the evolving or linear nature of time, which determines its arrow, is dependent on the A coefficient, so the quest for the best possible ‘atomic clock’ runs somewhat contrary to a full understanding of the physical properties of time.

Returning to the issue of how the second law of thermodynamics imposes the arrow of time, it is worth emphasizing again here that, also in Quantum Theory, the arrow of time is ultimately a consequence of this law. The non-perturbative introduction of the radiation field is merely an intermediate step, enabling the second law to play its full role. Time, of course, is a property of the Universe, not of a single atom, so a Universe containing many atoms is required in order to apply the present ideas. This raises the question: how is the arrow of time defined during the period between the Big Bang and the birth of the very first atoms? Obviously, one requires other phenomena to establish an arrow of time in an environment entirely devoid of atoms, but still subject to Quantum processes.

For a long time, philosophers have described the dual nature of time, which is both cyclic (or repetitive) and evolving (or linear). Sometimes, even these two aspects were considered as alternative descriptions. The formalism of Quantum Mechanics supplemented by the introduction of both the A and the B coefficients, provides a description of this duality based on sound scientific principles, showing that, actually, the two aspects are not alternative but complementary.