The quantum theory of time, the block universe, and human experience.

Advances in our understanding of the physical universe have dramatically affected how we view ourselves. Right at the core of all modern thinking about the universe is the assumption that dynamics is an elemental feature that exists without question. However, ongoing research into the quantum nature of time is challenging this view: my recently introduced quantum theory of time suggests that dynamics may be a phenomenological consequence of a fundamental violation of time reversal symmetry. I show here that there is consistency between the new theory and the block universe view. I also discuss the new theory in relation to the human experience of existing in the present moment, able to reflect on the past and contemplate a future that is yet to happen.This article is part of a discussion meeting issue 'Foundations of quantum mechanics and their impact on contemporary society'.

Introduction

Advances in science have had a profound impact on how we view ourselves. In the 1500s, the Copernican model of the solar system overturned the Ptolemaic model and with it the notion that human activity was a centrepiece of the cosmos. Over the course of the next three centuries, the development of classical mechanics led to the view of a clockwork universe that operated deterministically. The extent of the determinism was extreme to the point considered by Laplace where a being (i.e. Laplace’s demon) with sufficient resources could, in principle, predict the entire future of the universe from knowledge of its present state [1]. The notion that we, as a part of the universe, can act freely was challenged as a result. A new challenge emerged a century ago with Minkowski’s space–time [2] as it provided support for viewing the universe as a four-dimensional space–time block that exists as one entity. In this view, called the block universe (or eternalism in philosophical discussions), there is no basis for singling out a present time that separates the past from the future because all times coexist with equal status. As such, it has profound implications for how we view our experience of existing in the present moment, able to reflect on the past and plan new actions in a variable future. Indeed, if all times coexist with equal status, the future is as fixed as the past and so humans can only act in compliance with the future; this directly conflicts with the notion that humans can act freely.

However, the discovery of quantum mechanics in the last century has left puzzling implications in its wake, particularly in relation to the status of local realism, which asserts that effects are local in the sense of propagating no faster than the speed of light and physical properties are real irrespective of whether they are measured or not [3-5]. The central issue is that quantum mechanics appears to allow for statistical effects that are instantaneous despite being distant from their causes, a situation that Einstein once referred to as ‘spooky actions at a distance’ [6]. There is a lack of consensus as to how we should interpret this situation. My own view is influenced by the debate on celestial motion from the 1500s which, to me, is analogous to the present situation. The retrograde motion of the outer planets that arises in the Ptolemaic model can be considered as a modern-day analogue of the instantaneous spooky action associated with quantum mechanics in the following way. The retrograde motion is not fundamental, in that it is not the actual motion of the planets, but rather it is how the motion of the planets appears from the perspective of the Earth, which has a special status of being the centrepiece in the Ptolemaic model. In the same manner, the instantaneous spooky action should be considered not as fundamental, but rather as a result of taking a particular perspective where humans (or certain other systems) are given the special status of being essentially independent of the rest of the universe. The special statuses and corresponding perspectives are removed on adopting the Copernican model and Brans’ fully causal model [7,8] in the historic and modern-day cases, respectively. Moreover, the retrograde motion is explained away as being apparent motion and the instantaneous spooky action is accounted for by a common cause [7,9], respectively. Further details of this argument can be found in appendix A. The point I make here is this. Although Laplacian determinism challenges the notion that humans can act freely, insisting humans are not shackled by Laplacian determinism does not actually clash with classical physics because the physical actions of any human can be mimicked by an appropriately programmed and sufficiently dexterous robot entirely in accord with classical physics. But now, following the discovery of quantum mechanics, we find that insisting human actions are essentially independent of the rest of the universe has an apparent physical consequence in the appearance of instantaneous spooky action. The fact that this consequence can be placed in analogy with the apparent retrograde motion associated with the Ptolemaic model undermines the argument for human independence. As such, it poses a new challenge to the notion that humans can act freely.

The tendency for scientific advances to reinforce the case for a mechanistic model of ourselves shows no sign of abatement. In the following, I discuss how my recently introduced quantum theory of time [10,11] maintains this trend. In particular, the new theory involves a set of states, each one of which represents a physical system such as a galaxy as existing, in the case of time reversal symmetry, at one specific time. However, in the case of a violation of time reversal symmetry (T violation) of the kind observed in particle physics [12,13], the set of states represent the galaxy as existing at a sequence of increasing times. Each state provides ontological support for our subjective perception of existing in the present and the ordered sequence of states provides ontological support for our perception of progressing through time from the past to the future. The theory may thus appear to be at odds with the block universe for which events at all times coexist as one entity. However, I argue below that the new theory is, in fact, consistent with the block universe view. I show that the ordered sequence of states provides a physical basis for some of our subjective notions about time to be incorporated into the block universe view. It resolves, therefore, the open problem of how to square the human perspective of time within the block universe [14]. Yet despite this, it offers no escape from determinism and the mechanistic human.

The organization of the remainder of this paper is as follows. I rework the original development of the new theory [10] into a set of four basic principles in §2 and summarize the main results of their implementation in §3. Next, in §4, I interpret the new theory in terms of the block universe and then, in §5, I discuss the implications for how we experience time. I end with a conclusion.

Principles of the quantum theory of time

The development of the new theory [10,11] can be distilled into four basic principles, each of which originates from a specific observation. The first observation is that the accepted convention in physics is that dynamics is assumed to be an elemental part of nature—as existing without question—and is incorporated into physical theories through conservation laws and equations of motion. However, this convention treats time differently to space because, in essence, systems evolve and conservation laws apply over time and not space. This asymmetric treatment is at odds with the structure of Minkowski’s space–time which expresses an equivalence between time and space [2]. If this time–space asymmetry was found to be phenomenological in origin, we might gain a better understanding of the nature of dynamics and time itself. Accordingly, we begin with a representation of the quantum state of a system that treats time and space symmetrically by adopting the following:

Principle 2.1.

The physically realizable states of a system have the same construction in both time and space. Any differences between space and time, such as dynamics and conservation laws, emerge phenomenologically.

The important point here is that it allows a physical system to be represented as being localized in both space and time. The kinds of systems being considered could be any system describable in conventional quantum mechanics. Allowing localization in time rules out the possibility of incorporating equations of motion and conservations laws at a fundamental level of the theory.

Next, we note that the generators of translations in time and space are given by the Hamiltonian and the momentum operators , respectively. They are distinguished by the discrete symmetries of charge conjugation (C), parity inversion (P) and time reversal (T) in the sense that only the temporal generator expresses violations of the symmetries. In fact, the violations are only seen in the time evolution of various particle decays and thus in translations over time. This suggests that the discrete symmetry violations may provide a phenomenological basis for the kind of time–space asymmetry needed for dynamics to emerge in accord with principle 2.1. In particular, if represents the Hamiltonian for evolving into the future, then the Hamiltonian for the time-reversed evolution into the past is where is Wigner’s time reversal operator [15]. For brevity, we use a special symbol for each version of the Hamiltonian, namely where the subscripts ‘F’ and ‘B’ represent the ‘forwards’ (or positive t) and ‘backwards’ (negative t) directions of time evolution, respectively. T violation is represented by This gives rise to a seemingly ambiguous situation where there are twice as many unitary time translation operators as are needed. The following principle provides the necessary clarification of the role of each type of operator; its justification can be found in Vaccaro [10].

Principle 2.2.

Physical time evolution is represented by the operators[1] and for the forward (positive t) and backward (negative t) directions of time, respectively, where δt>0 represents an interval of time t. The operators and represent the mathematical inverse operations of ‘unwinding’ or ‘backtracking’ the evolution represented by and , respectively.

The situation for spatial translations involves no such subtlety. The operator representing a translation by a distance δr in the direction of unit vector n is given by ; the corresponding operator for a translation in the reverse direction is simply where is the parity inversion operator. Note that these two spatial translation operators are inverses of each other.

The third observation is about Feynman’s sum-over-paths (or path integral) formalism [16]. Although the formalism may be considered a mathematical tool for calculating probability amplitudes, Feynman’s demonstration that it gives the origin of Hamilton’s least action principle in classical mechanics suggests that the associated interference between multiple paths has ontological significance even to the extent of underlying the large-scale classical structure of the universe. As such, sums over paths should feature at a fundamental level in quantum theory. Paths over space are traversed by application of the spatial translation operators and for where the values of j, k and ℓ are randomly chosen from the set {1,−1}. Note that the spatial translation operators are mutually commuting. Paths over time are traversed by application of the time evolution operators and for δt>0 according to principle 2.2, and hence paths with reversals in direction contain both types of time evolution operator. More reversals implies more instances of each kind of evolution operator. A particularly important case is where and are non-commuting because then paths over time become quite distinct from paths over space. This phenomenological asymmetry between time and space is the kind we seek, and hence we adopt the following principle.

Principle 2.3.

Let W represent a temporal or spatial axis and w represent a coordinate value on it. Let |w0〉w be a ket that is extremely localized near w=w0 to the extent that it is not representative of any physically realizable state[2] . We express physically realizable kets |ϕ(w0,σ)〉w in terms of |w0〉w as follows. Imagine translating |w0〉w over all possible paths along W where each path comprises N steps; each step is a translation by the interval in either ±w direction, and σ is a width parameter. The translation of |w0〉 over each path will result in an extremely localized ket |w′〉w at a corresponding final coordinate value w=w′. The ket |ϕ(w0,σ)〉w is defined to be the superposition of all such translated kets according to where is the discrete symmetry operator associated with W (i.e. either the time reversal or the parity inversion operator), and is the generator of translations along W in the ±w directions, respectively. The right-hand side of equation (2.4) is called a ‘quantum virtual path’ (QVP) over W; it expresses the ket |ϕ(w0,σ)〉w in terms of a mathematical construction that is regarded in the theory as being fundamental.

QVPs extend Feynman’s sum-over-paths formalism [16] in two different ways. The first is that, whereas each of Feynman’s paths are, in general, possible trajectories in space and time, here each path represents a sequence of virtual displacements in either space or time. Virtual displacements arise in analytical mechanics when discussing constraints on motion [17]. A QVP is the superposition of all possible paths of this kind (where the adjective ‘quantum’ is used in the same sense as in the label ‘quantum walks’ [18]). The second is that, whereas Feynman used sums over paths as a means for calculating the transition amplitude for a process with respect to an initial and a final state, here a QVP is associated with the quantum state of a system itself.

The final observation that we need to construct the new theory is that there are expected to be fundamental limits to the degree to which intervals in time and space can be probed. For example, quantum gravity is expected to limit the resolution to the Plank time ≈10−44 s and the Planck length ≈10−35 m [19-21]. This implies that QVPs over the W axis with step sizes δw in equation (2.3) that are smaller than the corresponding fundamental limit δwmin should be accorded the same physical status in representing the state of the system. Even though the expected limits are extremely small, they allow for an unbounded number of QVPs of equivalent physical status with δw<δwmin. To account for this, we adopt the following principle.

Principle 2.4.

There is a lower limit to the resolution of intervals in space and time. QVPs with step sizes smaller than this limit have an equal physical status in representing the state of the system. The set of all QVPs with equal physical status is given by where Nw is the smallest value of N in equation (2.3) that gives δw<δwmin. Each QVP in is associated with identical values of the physical parameters σ and w0 and is distinguished mathematically by a unique value of the physically inaccessible index N. The set itself is, therefore, a representation of the state of the system characterized by the parameters w0 and σ.

The underlying concept here is closely related to the construction of the set of real numbers: representing a state by the set of QVPs with equivalent physical status is analogous to a real number being represented by an equivalence class of Cauchy sequences of rational numbers [22].

Overview of the main results

We turn now to the main results that follow from implementing the four Principles; further details can be found in the original work [10]. Consider first the situation where there is only one generator of translations over the generic axis W, i.e. where in equation (2.5). Making use of the result shows that every QVP in is physically indistinguishable from the limit ket .

The one-generator situation occurs for the one-dimensional spatial case where corresponds to the momentum operator in the x direction, say. In that case, we replace the generic label ϕ with ψ and note that |w0〉w corresponds to the position eigenstate |x0〉x, the set corresponds to and Nx is the smallest value of N that gives , where δxmin is the fundamental spatial resolution limit. Each QVP in is found to be physically indistinguishable from the Gaussian-distributed limit ket The set is redundant in the sense that it contains infinitely many QVPs that are equally able to represent the state of an object that is located near x=x0 within a region of order σ. The set itself is, therefore, a representation of the one-dimensional spatial state of the object in accord with principle 2.4.

The one-generator situation also occurs for QVPs over time in the case of T symmetry where corresponds to , and hence an equivalent situation results. Using Υ instead of ϕ as the label gives the set corresponding to equation (2.6) as where Nt is the smallest value of N that gives where δtmin is the fundamental temporal resolution limit. Just as in the spatial case, each ket in is found to be physically indistinguishable from the Gaussian-distributed limit ket: Two comments are in order here. First, the ket |t〉t is related to a fixed ket |t0〉t by time translation, i.e. . Second, the fixed ket |t0〉t is assumed to be well-defined in time in the sense that translating it in time by a small amount yields, essentially, an orthogonal state, namely the value of t〈t|t0〉t is negligible except near t=t0; this implies that |t0〉t has a correspondingly very broad energy distribution.

The set represents the galaxy as existing temporally only for a duration of the order of σ around the time t0 in the same way that represents it as existing spatially in a region of order σ near the spatial point x0. Clearly, these states have the same construction in time as in space in accord with principle 2.1. As a consequence, the state represents the violation of the conservation of mass and energy. There is also no equation of motion inherent in the theory to represent time evolution of the galaxy over extended periods of time. The study of states that are localized in time like this has a history dating back to Stueckelberg in the 1940s and Feynman, Nambu and Schwinger in the 1950s [23]. However, in contrast with that here, these previous studies incorporated dynamical equations as an integral part of their approaches and inherently assumed time evolution.

Next, consider the situation where there are two distinct generators, i.e. . This occurs for QVPs over time when T violation is present, i.e. . Let and in equation (2.5) correspond to and , respectively. We assume the commutator of the two versions of the Hamiltonians to be of the simple form The resulting QVPs depend on the physical parameter λ in addition to t0 and σ, and hence we need to include it in expressions accordingly. Thus, in place of equation (3.3), we now have The non-commutativity of the time evolution operators and leads to destructive interference between the virtual paths to the extent that the QVPs in are given approximately, on coarse-graining over time, by Vaccaro [10] where

for 2π<θ<4π. Each term on the right-hand side of equation (3.7) represents a modified QVP that has different contributions of time translation by and in accord with principle 2.2. The operators and are the corresponding weighted averages of and and represent the phenomenological Hamiltonians that would be observed within the galaxy. They can be thought of as the effective generators of the net time translation in each term on the right-hand side of equation (3.7). The parameter tc represents the corresponding net translation in time that would be observed within the galaxy using conventional clocks.[3] The particular value θ=2.23π corresponds, approximately, to each term on the right-hand side of equation (3.7) being associated with the minimum uncertainty in time that is allowed by the minimum possible uncertainty in energy.[4]

The ket |Υ(t0,σ,λ)〉t in equation (3.7) represents the galaxy existing in a superposition of the two times t0±tc which reflects the symmetry inherent in the definition of QVPs in equation (2.4). Selecting increasing values of N gives a sequence of kets |Υ(t0,σ,λ)〉t in associated, according to equation (3.11), with increasing values of tc. One take of this situation, based on the Feynman–Stueckelberg interpretation of antiparticles as particles travelling backward in time, is that the galaxy is matter-dominated at one of the times, say t0+tc, and antimatter-dominated at the other time, t0−tc. Bi-directional time evolution like this has been explored previously by Carroll, Barbour and co-workers [24,25]. To see what it means here, consider a mechanical clock based on the escapement mechanism with hands that rotate clockwise with respect to increasing values of tc and constructed from matter in the matter version of the galaxy at time t0+tc; a little thought will reveal that a corresponding clock device in the antimatter version of the galaxy at time t0−tc, that is equivalent in every respect (including the shape of the teeth on the escape wheel) except being constructed from antimatter, will have hands that also rotate clockwise with increasing values of tc. This implies that if both clocks are synchronized at one particular value of tc, they will remain in synchrony for larger values of tc. As the value of the parameter t0 that labels the ket |t0〉 is arbitrary, we set for convenience. We assume all clocks, irrespective of where they are located, are synchronized to display t, which hereafter we refer to as the clock time. Also, for brevity, the location of the galaxy will only be quoted explicitly for positive values of time, leaving the corresponding situation for negative values implicit, unless clarity warrants otherwise. The values of tc that correspond to all the kets in is found from equations (3.7) and (3.11) to cover an unbounded range starting from the minimum value of . Interpreting as a state in accord with principle 2.4 implies, therefore, that the galaxy exists for all times except in the interval (−tc,min,tc,min). Outside this interval the galaxy is represented as existing essentially continuously and, in this sense, conservation of mass arises as a property of . The dynamics of the galaxy follows the Schrödinger equation in the sense that and and hence the first and second terms on the right-hand side of equation (3.7) represent time evolution in the backward and forward directions of time, respectively. Moreover, the set is ordered because, in line with principle 2.2, is related to for t′c>t by a net time evolution according to[5] for δtc=t′c−tc, whereas the relation gives the rewinding of equation (3.15). This justifies taking the ket |Υ(t0,σ,λ)〉t to be correspondingly more evolved in time compared to |Υ(t0,σ,λ)〉t for n>0. All of these results taken together show that the new theory offers a framework that gives the origin of dynamics.

Consistency with the block universe

We are now in a position where we can interpret the new theory in terms of the block universe. For this, we restrict our attention to T violation and the temporal case. An individual ket such as |Υ(t0,σ,λ)〉t∈ describes the galaxy when its internal clock time is tc according to equation (3.11), and hence it could be regarded as a conditional state of the galaxy for a given value of tc. The full or unconditional state of the galaxy, according to principle 2.4, is the whole set . In particular, as all elements in have equal physical status, there is no basis for choosing one element as representing the ‘present’ state of the galaxy such that it divides the remaining elements into those representing ‘past’ or ‘future’ times. Moreover, represents the galaxy evolving according to the phenomenological Hamiltonian associated with the corresponding direction of time as described by equation (3.7). As such, the dynamics of the galaxy is fixed for all times apart for the small interval (−tc,min,tc,min). In this respect, the new theory is consistent with the principles of the block universe view.

The new theory, however, includes an additional feature. The direction of time in the conventional block universe is the one associated with increasing entropy. Conventionally, only time-symmetric laws are considered, and hence any time asymmetry in entropy is necessarily associated with asymmetric temporal boundary conditions, i.e. with the universe having low entropy at one extreme time and high entropy at another [26]. By contrast, the set has an inherent temporal order such that an element associated with larger values of N is more evolved in time, as discussed above. This implies that if we pick one element of , say |Υ(t0,σ,λ)〉t, it would contain evidence of having evolved from |Υ(t0,σ,λ)〉t for N′tc where, in general, each conditional state contains physical evidence of the prior time evolution represented by the conditional states |Υ(t0,σ,λ)〉t for t′c<tc where from equation (3.11).

The direction of time here is not set by the entropic nature of the galaxy, but rather it is due to the degree of time evolution in accord with principle 2.2 and this, in turn, determines the direction in which entropy increases. As a result, the new theory allows the block universe view to be extended, accordingly, to one containing an inherent direction of time.

Implications for humans

The new theory offers some support for our subjective notions about time. In particular, although not evident from the coarse-grained approximation given in equation (3.7), the conditional state |Υ(t0,σ,λ)〉t represents the galaxy in coordinate time t as being spread over a small interval around t=±tc. In fact, a more accurate expression for |Υ(t0,σ,λ)〉t is a superposition of two Gaussian-distributed kets centred on the times t=±tc and each one with a variance in time of where λ and θ are given by equations (3.5) and (3.10), respectively [10]. The fact that the galaxy is represented by |Υ(t0,σ,λ)〉t as localized in time is a sufficient objective basis to support the subjective perception of existing at a ‘present’ moment at time tc. Moreover, in the previous section we found that each conditional state |Υ(t0,σ,λ)〉t contains, in general, physical evidence of time evolution prior to tc, and this provides an objective basis for contemplative human observers within the galaxy to perceive recalling the ‘past’. In other words, whenever the galaxy exists as represented by |Υ(t0,σ,λ)〉t, contemplative human observers could perceive the present moment and recall the past. As the conditional state |Υ(t0,σ,λ)〉t represents the galaxy existing only for a small duration around tc, the subjective perception of the present moment and recalling the past is likewise confined to the same region of time. However, as all elements of have equal physical status, if one element has this representation, then they all do equally! The only option, then, is for the set to be an objective basis for humans to perceive the present moment and recall the past at all associated times tc.[6]

There is a further implication for us in the distribution of the conditional states |Υ(t0,σ,λ)〉t over time. The increase in tc for successive values of N is given by which is less than  [10], where δtmin is the temporal resolution limit, and hence the set of values of the clock time tc are physically indistinguishable from a continuous range. The fact that the spread in time Δtc given by equation (5.1) is independent of N, whereas decreases as N increases implies that the conditional states |Υ(t0,σ,λ)〉t for sufficiently large, successive values of N will overlap in time. A given clock time tc will, therefore, be associated with the ordered collection of conditional states such as {|Υ(t0,σ,λ)〉t:|tc−t′c|≤Δtc} where . In other words, any particular clock time value tc is associated with an ordered collection of conditional states. To make a comparison with human experience, note that the timescale associated with conscious awareness is of the order of a second [27], which is many orders of magnitude greater than the intervals such as Δt considered here. The subjective awareness of time is associated, therefore, with a set of many conditional states |Υ(t0,σ,λ)〉t that overlap in time and are ordered in terms of increasing clock time. These time-ordered conditional states of the galaxy carry a description of time-ordered neurological effects in the humans within it, and this provides an objective basis for the subjective perception of a ‘flow’ or ‘passage’ of time.

Finally, there is also a severe constraint for us. The new theory is deterministic in the sense that it obeys Hamiltonian dynamics and hence it does not allow us to act freely nor to have an ability to determine the future, despite our strong subjective perceptions to the contrary. In short, it offers us no escape from determinism.

Conclusion

Quantum mechanics has spawned the development of a myriad of advanced technologies that drive the modern western way of life. However, the foundations of quantum mechanics offer something more, a reason to question how we view ourselves within an objective landscape. Indeed, the test of local realism formulated by Bell [5] appears to be a new challenge to the notion that humans can act freely.

In this paper, we explored the implications of the quantum nature of time for humans. The structure of the recently introduced quantum theory of time [10] was distilled into four basic principles and the main results that arise from their implication was also briefly reviewed. An important feature of the new theory is that dynamics and time evolution are not assumed to be part of its structure, rather they emerge phenomenologically due to T violation. The state of a physical system over time is given by a collection of conditional states, where each conditional state represents the system at a specific value of clock time.

We found the new theory is consistent with the block universe in that the conditional states are fixed by unitary evolution and that no one conditional state can be singled out as representing the present, and divide other conditional states into sets representing the past and the future. However, the new theory has additional features. One is that each conditional state is localized in time around a mean clock time value and hence it can represent a human having the subjective perception of the present moment. Another is that the collection of conditional states has a natural ordering in terms of the degree of time evolution, and as a result, each conditional state contains, in general, evidence of past evolution. This supports the subjective perception of being able to recall the past and contemplate the future. It also provides an objective basis for the subjective perception of a flow or passage of time. As such, it offers an alternative to Price’s view that the subjective passage of time does not have an objective basis [14]. In short, the collection of conditional states can represent a human at every clock time as having the subjective perception of the present moment with memory of the past and awareness of the passage of time. These results extend the block universe view. Finally, the natural ordering due to T violation suggests a connection with causality [28,29] but this is yet to be explored.