Parashat Tzav 5782 — 03/19/2022
Beginning with Bereishit 5781 (17 October 2020) we embarked on a new format. We will be considering Rambam’s (Maimonides’) great philosophical work Moreh Nevukim (Guide for the Perplexed) in the light of the knowledge of Vedic Science as expounded by Maharishi Mahesh Yogi. The individual essays will therefore not necessarily have anything to do with the weekly Torah portion, although certainly there will be plenty of references to the Torah, the rest of the Bible, and to the Rabbinic literature. For Bereishit we described the project. The next four parshiyyot, Noach through Chayei Sarah, laid out a foundational understanding of Vedic Science, to the degree I am capable of doing so. Beginning with Toledot we started examining Moreh Nevukim.
Vayikra 6:1-8:36
This week I would like to look at wholeness in quantum mechanics from two perspectives. The first has to do with the Law of Least Action, and the second one has to do with the Unified Field. The two are actually intimately related, as we will see at the end of the exposition.
Let’s start with a single particle traveling in one dimension. The state of the particle can be described by two numbers: its position and its momentum (essentially its speed, plus = left-to-right or minus = right-to-left). We can draw two axes on a paper, the horizontal axis representing the particle’s position and the vertical axis representing its momentum. Then each point in this abstract space (we call it “phase space”) represents the complete state of our one-particle system. Based on the current state of the system and the forces acting on the particle, we can calculate where the particle will be and how it will be moving in the “next” moment of time. Thus, starting at point A in phase space we can calculate our way to any point B, tracing out the path through phase space (i.e. the sequence of position/momentum values that connect A and B). This is an inherently local perspective – the “next” position/momentum depends only on the current position/momentum and the forces acting on the particle at that point in phase space.
There is an elegant extension of this formulation that allows it to be applied to virtually any system, be it a system of particles moving in three-dimensional space, or a field that fills all space. Needless to say, the “phase space” for this larger system may have many, many more than two dimensions, up to an infinite number of dimensions. (For example, a single particle moving in 3-dimensional space will have a phase space of 6 dimensions – the three spatial dimensions, x. y and z, and the components of the momentum along the same three dimensions.) The basis of this formulation is a function called the Lagrangian, which has a term analogous to kinetic energy (“energy of motion”) that depends on the momentum of the system at each point in phase space, and another term which depends on the potential energy, which depends on the position values of the system at each point in phase space (the “forces acting on the system” are calculated from the potential energy). Through an elegant set of equations (Hamilton’s Equations) the Lagrangian can be used to step our way from A to B as described in the previous paragraph.
There is another way to use the Lagrangian to get from point A to point B. If we take any random path through phase space from point A to point B, we can calculate the Lagrangian at each point. If we add up the values of the Langrangian from all points along the path we get a number called the action. There are, of course, an infinite number of possible paths from A to B, and each one has an action number associated with it. Nature does not appear to take all paths however – it takes (classically) only one particular path. That path is the path of least action, In other words, it is as if nature computes all possibilities simultaneously and chooses one that is optimal in some way. How does nature do this?
The answer was developed in 1948 by Richard Feynman and is called the path-integral approach. The basis of this approach is a phenomenon called interference. When two waves impinge on the same point in space at the same time, they can be in sync with one another or not. In sync means that the tops of the waves arrive at the same time and the troughs also arrive at the same time. The net effect is that the total amplitude of the wave is the sum of the amplitudes of the individual waves. (Since the intensity of a wave is proportional to the square of the amplitude, if the two waves are of equal amplitude, the total amplitude will be multiplied by 2, but the total intensity will be multiplied by 2 squared = 4. This is the basis of the superradiance effect, where only the square root of 1% of a population is needed to create peaceful and coherent effects for the whole population.) When waves are in sync with one another we have what is called constructive interference. The in-sync waves are called in phase with one another.
If the two waves are completely out of sync, with the crest of one arriving along with the trough of the other and vice versa, the net effect is that the two waves cancel each other out and the resulting amplitude (and intensity) is zero. This is called destructive interference. We say these waves are 180° out of phase with one another. If we have many waves with the phases distributed evenly from 0 to 360° then every individual wave will find it’s 180°-out-of-phase “mate” and the result will be zero.
Now let us return to our paths and their action. Feynman posited that attached to each path is a wave, and the number of wavelengths from point A to point B is given by the action for that path divided by Planck’s constant, h (=6.62607004 × 10-34 Joule-sec). This is an extremely small number, and therefore action / h is an extremely large number of waves for any macroscopic (human-scale) system. For an atomic-scale system, however, any particular path (say an electron orbiting a nucleus) there may be only a very few waves corresponding to the action for that path.
Now the path that we “see” in nature is the path of least action – that path for which the action, and therefore the number of waves, is minimized. Now the interesting thing about a minimum is that at the minimum, things are flat. We know this from driving in hilly country. We go down, down, down into a valley. When we come to the bottom, we are level and then we start going up, up, up the other side. The minimum is the junction point between going down and going up, but it, itself, is neither going down nor going up. This is true even if the bottom of our valley is but a single point – at that one point the valley is flat. If we go a little bit to one side or the other around the minimum, the height does not change appreciably. In contrast to this, if we go a little bit to one side or the other of any point on the up- or down-slope on the sides of the valley, the height does change appreciably.
Now let’s put all the pieces together. The path of least action is the path with the minimum action of all the possible paths between point A and point B. Therefore, if we deviate slightly from this path, the action doesn’t change very much, and consequently the number of wavelengths for paths near to the path of least action will be almost the same. That means that all these waves are coherent with one another, in phase with one another, and we get constructive interference. The total intensity of the waves from the paths around and including the path of least action is very high – and this is the path we see nature taking.
In contrast, at some distance from the path of least action, if we deviate slightly from any chosen path then action will deviate quite a bit. Thus, a bundle of paths some distance will have wildly varying numbers of wavelengths arriving at point B and there will be destructive interference. The only thing left standing is the path of least action.
I have only sketched the outline of this approach; if you want a more rigorous and detailed explanation, please follow the links I gave above.
Let’s take a step back. We started out trying to understand why nature takes the path of least action. Now we find that nature doesn’t only take the path of least action – in fact it takes all paths, all of the time, and winnows out the path of least action for our eyes only! How non-local is that! Nature computes the action for all paths and uses all paths, all at once, and does so for the entire universe, a system with infinite dimensions and infinite numbers of possible paths. Our notion that one particle at one point causes something to happen at that point, seems almost ludicrous.
In fact, there are much subtler mechanisms of causality that might come into play. Under our old system of thinking, if I’m sitting on a pillow meditating, I am not flying through the air. Under the old paradigm, it would require a force to keep me hovering even a millimeter over my pillow in defiance of gravity. With our new principle, we see that in reality I am already hovering over my pillow, and it’s just because of destructive interference that I can’t see this (nor can anyone else!). But suppose in my meditation, from a very subtle level, I could adjust the phase of some of the “hovering” waves so they were coherent. Then “hovering” would be the visible behavior and sitting would not. Please note that I am not recommending this as a technique, rather I am just suggesting how such a technique might work on a subtle level.
I hope to wrap up this long excursus into the nature of causality next week, with a discussion of Unified Field theory.
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Commentary by Steve Sufian
Parashat Tzav
In the previous parshah, Vayikra, Gd called to Moses to tell Aaron and his sons, the priests, the nature of the offerings they will make. In this parshah, Tzav, Gd commands Moses to tell Aaron and his sons their rights and their responsibilities regarding the offerings and the eternal fire into which the offerings are made.
The symbolism here is very sweet: in order for the fire to be eternal, to not go out, it needs to be fed each morning with fresh wood. The eternal fire symbolizes the relation between Gd and humanity; for it to be eternal it needs to be fed each day with our right actions so we can experience the Full Restoration of our awareness to Oneness with Gd, the Eternal, the One.
Similarly, the fire symbolizes the Fire of our own soul, which guides us to act lovingly so our actions are good actions, our actions draw us near to Gd and also near to all the expressions, Creation, of Gd: our family, friends, neighbors, strangers, trees, plants, rivers, stones — all the expressions of Gd.
And in order for this fire to be kept burning, for our soul to be kept interacting with Gd and with Gd’s world, we need to make offerings, not only every morning as with wood for the eternal fire in the Tabernacle, but every moment — lest our soul withdraw from our personality, distance itself from Gd, if we fail to draw near to our soul by offering our good actions to Gd.
Baruch HaShem