Time and Energy Issues

In practice, there are many ideas about the nature of time, for example, is it absolute or relative, discrete or continuous, reversible or irreversible. Even within the development of the quantum model, different ideas about time have been proposed. Originally, in quantum mechanics, position [x] was considered as an operator, while time [t] was only treated as a variable, although this asymmetry was later considered to be problematic in terms of special relativity. Therefore, in Quantum Field Theory, position [x] and time [t] were both defined as variables, while the quantum field was described in terms of an operator. While the initial idea of time in quantum mechanics was possibly ambiguous, the development of the Planck scale led to the suggestion that time might also be discrete, although it did not initially affect the ‘arrow of time’. However, later, within the development of quantum field theory, the idea of time was subject to further speculation based on the conceptual nature of virtual particles, described as transient quantum fluctuations. One such speculative idea was the Transactional interpretation, which possibly makes so many untestable assumptions that this review will not even try to summarise the details.

Note: In a broad context, classical thermodynamics generally proceeds on the idea that time is irreversible in the sense that a closed system will always decay towards its lowest energy equilibrium state. However, whether any system in the universe, including itself, is truly closed may be debated, especially at the quantum level.

In a somewhat contradictory position to classical thermodynamics, classical physics did not necessarily prohibit the reversing of time as its equations of motion were essentially reversible, at least in concept. However, this fact does not necessarily lead to the conclusion that time itself is reversible as one is simply a mathematical abstraction, while the arrow of time might be seen as a physical reality. However, at the start of the 20^th century, Herman Minkowski proposed that space-time could be merged into 4D spacetime on the basis of special relativity. Again, this review might question whether this was only another mathematical abstraction for none of the ideas, briefly outlined, appear to provide any causal explanation of time, although one is suggested below.

Note: Within the context of a wave-like model of everything, time [t] might be seen as the reciprocal of wave frequency [f=1/t], while the measure of space [x] is defined by wavelength [λ=x]. If, at the quantum level, everything has a wave-like nature, then the granularity of time might correspond to the highest frequency [f_P] in the quantum universe. Likewise, on the assumption that these waves are subject to a finite propagation limit, e.g. [c], at least in the present era, then the granularity of space would correspond to the shortest wavelength related to the highest frequency, i.e. [λ_P=c/f_P], where the suffix [P] denotes the Planck scale.

Again, we might need to continue questioning assumptions that appear speculative in the absence of any empirical evidence. In this context, we might recognise that science has offered up many descriptions of energy, which classical physics quantified in terms of mass [kg]. As a broad generalisation, total energy [E_T] is a combination of potential [E_P] and kinetic [E_K] energy, where rest mass may only be a manifestation of both dependent on velocity. As indicated, energy is quantified in terms of mass [kg] irrespective of whether discussing a potential field, kinetic motion or a body at rest [E₀]. However, mass units of energy also apply to radiation [E_R], where no rest mass exists, as illustrated in [1].

[1]   

It might be useful to further illustrate some of the different formulations of energy, where mass [kg] might first have some logical meaning in terms of classical particle physics. However, the idea of mass becomes increasingly ambiguous in the quantum model, if the substance of the particle is only a manifestation of energy-density associated with a waveform, especially in the context of radiation energy.

[2]   

But, how might we explain energy without mass?

We might initially consider this question in terms of radiation energy, although it is unclear that [E_R=hf] provides any causal explanation of how this energy propagates with velocity [c], other than as defined by electromagnetic theory. However, without being too specific about details, the potential energy [E_P] of a wave, at any single point in space and time, might be modelled in terms of Hooke’s law, as simply represented in [3].

[3]     

For the purposes of this discussion, we might equate [A] to the displacement amplitude, where [k] is assumed to be a constant of that which is being displaced, e.g. coiled spring or quantum field. If so, we might recognise that amplitude [A] has to change as a function of time [t] and frequency [f], possibly as described in [4].

[4]     

So, how might we equate [3] and [4] with a photon of energy?

Although we have no knowledge of the structure of a photon, we might assume that the energy inferred by [E=hf] has some space-time granularity, i.e. it is a quantum of energy, such that it does not exist at a single point in space, as per [3]. If so, any practical measure of this quanta of energy has to be an aggregation of A(t) within a volume [V] of space, subject to time. While we do not necessarily understand the exact details of the energy-density [ρ] within this quanta, we might assume an average energy-density and volume [ρV]. However, while this outline might appear to be describing a photon quanta being particle-like, few would reject that it has wave-like properties, such that frequency [f] and amplitude [A] might have to be factored into the expression of this quanta of energy. As such, we might consider the implication of the equation in [5].

[5]     

However, at this point, we would like to avoid defining energy in terms of mass [kg], such that we might seek to convert mass [kg] into units of energy [Joules] by reversing Einstein’s equation [m=E/c²], such that the units of energy-density [ρ] become Joules/volume, as shown in [6].

[6]     

Note: It possibly needs to be highlighted that [A] represents an offset from a point of equilibrium, where potential energy is zero. While this oscillatory motion infers an internal exchange of energy between potential and kinetic forms, the variables [A] and [f] only reflect the potential of a point in space, which [ρV] aggregates within some finite volume. As such, [6] does not reflect the kinetic energy of the potential energy moving through space with velocity [v], which will be considered later in this discussion.

So, let us continue with the speculative idea that energy, whether as a particle or photon, has some form of wave structure, such that we might question whether Planck’s constant [h] is ‘hiding’ some of the wave-like parameters, which appears to be missing in Planck’s energy equation [E=hf] by referencing [6].

[7]     

At this point, we have simply highlighted a possible wave inference in [7], such that Planck’s constant [h] might be defined as shown in [8].

[8]     

As indicated earlier, [7] is only quantifying the potential energy within the energy quanta without making any reference to kinetic energy. Again, we might reference the idea of kinetic momentum [p], which we might consider based on the assumptions of relativity. However, if we start with Einstein’s energy equation [E=mc²], we might recognise that the scope of mass [m] has not been fully defined. For the purposes of this discussion, we might define two distinct types of mass related to the rest mass [m₀] when [v=0] and a kinetic mass [m_K] when [v → c], which can be used to clarify Einstein’s energy relationship, as shown in [9]

[9]     

Based on [9], we might rearrange as shown in [10].

[10]   

If we make reference back to [9], we see the revised form of Einstein’s equation [E=m_Kc²], where [m_K=m₀] only when velocity [v] is equal to zero. As such, we can rearrange [10], as shown in [11] to restore the equality to energy [E] and quantify one of the terms as equivalent to momentum [p].

[11]   

We can now apply [11] to a photon with zero rest mass where [v=c], such that energy [E] reduces to [pc], Alternatively, in the case of a stationary particle when [v=0], [11] reduces to rest mass energy [E=m₀c²], such that the full form of [11] only applies to a particle with velocity [v]. So, returning to the issue of a photon, we might revise the form of [11] based on the energy assumption in [7].

[12]   

Based on [12], we might see that photon momentum may still retain some wave-like characteristics, if the energy component [E] is equated to the expression in [7] leading to [12]. While the logic leading to [12] is far from rigorous, it is simply suggesting that energy of a photon may still be dependent on physical variables, i.e. energy-density [ρ] within volume [V] plus frequency [f] and amplitude [A]. However, even if we try to quantify the photon as a wave-pulse, where the known frequency [f=c/λ] might equate to the half-wavelength, there are still too many unknowns to quantified all the parameters in [12]. However, the main purpose of this exercise was to simply demonstrate the ‘possibility’ that some of the abstractions surrounding energy [E] and time [t] might be explained, and linked to causal mechanisms, if reference is made to wave-like variables. However, we might now reflect on the uncertainty of both position [x] and time [t], which then leads to uncertainty in both momentum [ρ] and energy [E].

[13]   

In [13], we see the basic form assumed to support the uncertainty principle, although at this stage, it is simply suggesting that any uncertainty in position [x] must imply an uncertainty in momentum [ρ]. However, if position [x] is actually associated with a distributed wavelength [λ], we might question why we would ever expect certainty in position [x]. We might now use [13] to make a similar argument with respect to energy [E] and time [t], as shown in [14].

[14]   

However, it is unclear why there should be any inherent uncertainty in time [t], if linked to frequency [f]. Of course, if uncertainty exists in momentum [p] based on the uncertainty in position [x] due to its dependency on wavelength [λ], it might still imply an indirect uncertainty in energy [E=pc] for the photon example discussed.

Note: For those interested in pursuing more details on the issues simply outlined in this discussion, reference might be made to a paper entitled: On Time in Quantum Physics.