In the previous blog post, we aimed at the questions which compel us to start a pursuit for a quantum theory of gravity. From now on, we will start paving the path for the discrete versions of spacetime. Till date, many ideas have been floated including mainstream continuum-based approaches such as string theory and loop quantum gravity and their discrete counterparts for instance causal set theory. Since decades, discrete versions could not grab public attention nor appreciated by a wider scientific community and as a result, these theories remain less incremental and face serious setbacks in their structural innovation despite of their intriguing predictability.

Causal theories provide a figurative approach to understand and model the universe around us. A glimpse of such a quantum universe is shown in the figure 2.1 [1]. Two different levels of structural hierarchy, employed in such causal universes, are:

  • higher-level structures,
  • lower-level structures.

Figure 2.1: Quantum universe in discrete causal theory (Credits: Benjamin Dribus [1])

In figure 2.1, the big open nodes and long edges are the higher-level structures and represent only a small portion of the quantum multiverse. Lower-level structures are drawn inside these open nodes with filled balls (or dots). Some of these nodes are connected through a line with other filled nodes while some are not. This reminds about the “ball-and-stick” molecular model. However, in physics, they seem not to be as complicated as they are in chemistry[1]. These simple structures are extremely powerful information systems and represent the even simpler classical universes or histories. Both the terms, universes and histories, are almost the same and widely used in causal theories. However, when we talk about ‘universes’, it evokes the idea of a ‘self-contained’ systems (independent of each other) while the term ‘histories’ depicts an ‘extended’ process and does not only indicate these nodes as the snapshots of underlying structure at a particular instance (or spacetime). Classical histories (or simply histories) are more descriptive in the context of discrete causal theories, so we will try to stick with this all the time.

At the moment, we are not talking about what these histories are required to model in conventional physics or what could be the size of these nodes? However, being the most fundamental structural unit of spacetime, these are required to describe and represent local processes such as particle interactions but it is expected, the size of these histories would be tens of orders of magnitude smaller than the interacting particles or any other objects involved in these processes.

Representation of Structures

The “ball-and-stick” or “node-and-edge” representation exists at both structural levels. However, these edges are not random but directed with initial and terminal nodes. These nodes are the events and the directed edges represent the cause-and-effect relation between these events. This is where causality comes in[2]. Such sets of events (or objects) are called directed sets[3]. Before proceeding, we will label the two-level structure in the following way:

  • Higher-level structures as the quantum structures,
  • Lower-level structures as the classical structures.

Many readers would be dubious about this labelling as it seems disturbing and unusual for many obvious reasons. Most trivial objection comes in if we think about the scales of natural phenomena. In physics, we use classical theory (such as Newtonian mechanics and Einstein’s general relativity) to study large objects which we are referring here as lower-level structures. On the other hand, quantum theory is an approach to study small-scale phenomena and we are labelling it as a higher-level structure. However, this labelling becomes obvious if we think about the Feynman’s path integral or path summation approach to quantum mechanics.

Richard Feynman, in 1948, presented his path integral formulation or sum-over-histories approach [2] which generalizes the action of classical mechanics. In classical theory, a particle can only have a unique trajectory from one point to another while in Feynman’s approach, it can take any path it wants out of infinitely possible trajectories. Each possible path is assigned a phase (or amplitude) in some space and then all the phases (or amplitudes) are summed up to yield the quantum amplitude and the probabilities of these amplitudes can then be measured. In Feynman’s own words [3]:

“The electron does anything it likes … It goes in any direction at any speed, forward and backward in time, however it likes, and then you add up the amplitudes and it gives you the wave-function.”

Another reason to think about this unusual labeling of structures is the supremacy of quantum theory. It is accepted that quantum theory is more fundamental than the classical, that, the classical world is built up from the quantum principles. In our designation, higher-level structures are responsible for making up the lower-level structures, nonetheless, this disagreement is not serious. To understand this, let’s try to comprehend what the word classical means.

  • Technically, it refers to a specific or unique permissible physical scenario[4], for instance, a unique trajectory to reach from one point to another. This is the lower-level structure. But in quantum world, we talk about all the possible permissible scenarios, henceforth, it evokes a higher-level structure.
  • Conventionally, anything, measured at sufficient large scales and low energies, is said to be classical. This was evident even before the development of quantum theory, when there was no concept of indivisible atom. However, in quantum theory, Bohr’s correspondence principle also provides a nontrivial justification to this definition. It states that [4] “the predictions of classical and quantum mechanics agree in the limit of large quantum numbers [the large scales].”

This comparison also clarifies an important issues in the understanding of classical and quantum theories. Conventionally, quantum theory is fundamental than its classical counterpart as the classical equations of motion are the approximations of quantum dynamical laws. However, technically, no theory is more fundamental than the other. Both are equally fundamental in their premises and represents different structural levels to explore the natural phenomena. But, on the structural basis, we can safely say that quantum (higher-level) structure is more complete than the classical (lower-level) structure because the quantum structure engirdles all the permissible physical scenarios while the classical only has unique or specific scenario.

How to Model a Picture with an Equation?

The way we have defined the hierarchical structure in the universe, it gives rise to a “structured configuration space”. Such a space represents a kinematic scheme and in case of discrete causal theories, these schemes are encoded by “sequential evolutionary processes”. A configuration space is a representation space of some underlying principle or phenomena and provides us all the possible states of a physical system. To understand the idea, let’s visit a very simple example from classical mechanics. The position of a particle in a system can be represented by three coordinates (x, y, z) or to describe its motion, we need different sets of the coordinates say (x_1, y_1, z_1) and (x_2, y_2, z_2) and it forms a function, say \psi(x_1, y_1, z_1, x_2, y_2, z_2), which lives in a six-dimensional configuration space. So, we can say, the configuration space represents the configuration or structure of the physical system[5].

It means, by adopting a generalized path summation approach, we can represent any quantum object with such a structured configuration space. And here comes the way to model a picture with an equation. To do is, it is necessary to find the contributions of all the paths in the configuration space and sum them up. In actual, to sum these contributions, it is needed to transform these structural points into some sort of mathematical objects and apply the suitable and pertinent mathematical operations. In the same fashion, the quantum amplitudes and other observables of the system can be found out. Such a scheme provides us the needed quantum-theoretic framework equipped with the predictive powers. In discrete causal theories, such an algebraic framework naturally arises from the underlying causal structure between the events[6] and this framework results in the dynamical laws for the theory. We now visit a particular example of such a dynamical law, the causal Schrodinger-type equation.

Causal Schrodinger-Type Equations

In non-relativistic quantum mechanics, time-dependent Schrodinger wave equation is

(2.1)   \begin{align*} i\hbar \frac{d\psi}{d t} = \hat{H} \psi.  \end{align*}

Here, \psi is an element of the Hilbert space and \hat{H} is the Hamiltonian operator. Since, it closely resembles with the classical heat equation, so its solutions are often dubbed as the wave functions or preferably state functions, since \psi represents the state of the system. Now I will only write the causal Schrodinger-type equation [1] and state what the symbols mean without going into the conceptual detail and derivation of the equation. Moreover, I will not show how this equation corresponds to the conventional Schrodinger equation. Anyhow, the discrete causal version of Schrodinger equation is[7]:

(2.2)   \begin{align*} \psi^{-}_{R;\theta} = \theta(r) \sum_{r^{-} \prec r} \psi^{-}_{R;\theta} (r^{-}).  \end{align*}

Here, \psi^{-}_{R;\theta} represents a generalized state function, and negative sign as a superscript indicates it is a past state function. The \theta(R) is the relation function and is analogous to the Lagrangian, rather than the Hamiltonian. Here, the relation function is defined for R which is a subobject in the configuration space and possesses a special kind of histories, the co-relative histories, for elements r^{-} and r. The symbol \prec in r^{-} \prec r reads as r^{-} precedes r or r^{-} is to the past of r^{-}. This relation encodes causality in the theory.

However, if this equation is true, it must have the necessary information to recover the continuum structure from the discrete spacetime, or in this case, quantum field theory and the Einstein’s field equations. But it is a long path to pave, so in the next blog post, we will try to understand what are metric recovery theorems and how we can recover the metric from the underlying discrete spacetime.

Note: PDF of this blog can be downloaded from here.

References

[1] Dribus, B. F. (2017). Discrete Causal Theory: Emergent Spacetime and the Causal Metric Hypothesis. In Discrete Causal Theory. https://doi.org/10.1007/978-3-319-50083-6

[2] Feynman, R. P., & Brown, L. M. (2005). Feynman’s Thesis: A New Approach to Quantum Theory. Retrieved from https://books.google.com.pk/books?id=5kowI7YgFbEC

[3] Lancaster, T., Blundell, S. J., & Blundell, S. (2014). Quantum Field Theory for the Gifted Amateur. Retrieved from https://books.google.com.pk/books?id=Y-0kAwAAQBAJ

[4] French, A. P., & Taylor, E. F. (1979). An Introduction to Quantum Physics. Retrieved from https://books.google.com.pk/books?id=HOCzTYEXpFIC


  1. We will figure out in future what the word complication means in this context.
  2. We cannot say, still, whether this representation truly depicts a Lorentz invariant causal structure or not. We will answer this once we develop a scheme to see how these nodes come into existence.
  3. I will explain this technically and in detail in the upcoming blogs.
  4. In discrete causal theories, these permissible scenarios are determined from the axioms of the theory and not from the classical equations of motion as we usually do in the electromagnetic theory or in general relativity. Histories satisfying these axioms are said to be permissible scenarios and treated on an equal footing. The histories, that do not satisfy axioms, are not even included in the theory. In contrast to continuum-based approaches, discrete causal theories do not distinguish between ‘on-shell’ and ‘off-shell’ scenarios.
  5. It is important to reiterate, configuration space is closely related to phase space or state space but not exactly the same.
  6. We will review this in detail in the future blog posts.
  7. This equation can also be treated as the discrete heat equation, for obvious reasons stated for the Schrodinger equation in non-relativistic quantum theory.
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