Update on 2015.09.07, 07 AM: I have effected a few corrections. In particular, I have made it explicit that the third quantity isn’t the strength of an independently existing third property, but merely a quantity that is registered when the two independent quantities are both being varied. Sorry about that. If the need be, I will simplify this discussion further and write another blog post clarifying such points, some time later.
The last time, I said that I am falling short on time these days. This shortfall, generally speaking, continues. However, it just so happens that I’ve essentially finished a unit each for both the UG courses by today. Therefore, I do have a bit of a breather for this week-end (only); I don’t have to dig into texts for lecture preparations this evening. (Also, it turns out that despite the accreditation-related overtime work, we aren’t working on Sundays, though that’s what I had mentioned the last time round). All in all, I can slip in a small note, and the title question seems right.
We often hear that the physical reality, according to physicists, is not the -dimensional reality that we perceive. Instead, it is supposed to be some -dimensional entity. For instance, we are told that space and time are not independent; that they form a -dimensional continuum. (One idea which then gets suggested is that space and time are physically inter-convertible—like iron and gold, for instance. (You mean to say you had never thought of it, before?)) But that’s only for the starters. There are string theorists who say that physical universe is -, -, or -dimensional.
What do physicists mean when they say that reality is -dimensional where ? Let’s try to understand their viewpoint with a simple example. … This being a brief post, we will not pursue all the relevant threads, even if important. … All that I want to touch upon here is just one simple—but often missed—point, via just one, simple, illustration.
Take a straight line, say of infinite length. Take a point on this line. Suppose that you can associate a physical object with this point. The object itself may have a finite extent. For example, the object may be extended over a small segment of this line. In such a case, we will associate, say the mid-point of the segment with this object.
Suppose this straight line, together with the -dimensionally spread-out object, defines a universe. That is a supposition; just accept that.
The -dimensional object, being physical, carries some physical properties (or attributes), denoted as . For example, for the usual -dimensional universe, each object may have some extent (which we have already seen above), as well as some mass (and therefore density), color, transmissivity, velocity, spinning rate, etc. Also, position from a chosen origin.
Since we live in a -dimensional universe, we have to apply some appropriate limiting processes to make sense of this -dimensional universe. This task is actually demanding, but for the sake of the mathematical simplicity of the resulting model, we will continue with a -dimensional universe.
So, coming back to the object and its properties, each property it possesses exists in a certain finite amount.
Suppose that the strength of each property depends on the position of the object in the universe. Thus, when the object is at the origin (any arbitrary point on the line chosen as the reference point), the property exists with the strength , the property exists with the strength , etc. In short the th property exists with a strength where is the position of the object in the universe (as measured from the arbitrarily selected origin.) Suppose the physicist knows (or chooses to consider) number of such properties.
For each of these number of properties, you could plot a graph of its strength at various positions in the universe.
To the physicist, what is important and interesting is not the fact that the object itself is only -dimensionally spread; it is: how the quantitative measures s of these properties s vary with the position . In other words, whether or not there is any co-variation that a given th property has with another th property, or not, and if yes, what is the nature of this co-variation.
If the variation in the th property has no relation (or functional dependence) to the th property, then the physicist declares these two properties to be independent of each other. (If they are dependent on each other, the physicist simply retains only one of these two properties in his basic or fundamental model of the universe; he declares the other as the derived quantity.)
Assuming that a set of some chosen properties such that they are independent of each other, his next quest is to find the nature of their functional dependence on position .
To this end, he considers two arbitrarily selected points, and . Suppose that his initial model has only three properties: , and . Suppose he experimentally measures their strengths at various positions .
While doing this experimentation, suppose he has the freedom to vary only one property at a time, keeping all others constant. Or, vary two properties simultaneously, while keeping all others constant. Etc. In short, he can vary combinations of properties.
By way of an analogy, you can think of a small box carrying a few on-off buttons and some readout boxes on it. Suppose that this box is mounted on a horizontal beam. You can freely move it in between two fixed points and . The `on-off’ buttons can be switched on or off independent of each other.
Suppose you put the first button in the `on’ position and keep the the rest of the buttons in the `off’ position. Then, suppose you move the box from the point to the point . The box is designed such that, if you do this particular trial, you will get a readout of how the property varied between the two points; its strength at various positions will be shown in a readout box . (During this particular trial, the other buttons are kept switched off, and so, the other readout boxes register zero).
Similarly, you can put another button into the `on’ position and the rest in the `off’ position, and you get another readout in the readout box .
Suppose you systematize your observations with the following notation: (i) when only the button is switched on (and all the other buttons are switched off), the property is seen to exist with units at the position and units at ; this readout is available in the box . (ii) When only the button is switched on (and all the other buttons are switched off), the property exists with units at and units at ; this readout is available in the box . So on and so forth.
Next, consider what happens when more than one switch is put in the `on’ position.
Suppose that the box carries only two switches, and both are put in the `on’ position. The reading for this combination is given in a third box: ; it refers to the variation that the box registers while moving on the horizontal beam. Let’s call the strengths registered in the third box, at and positions, as and , respectively; these refer to the combination (i.e. both the switches and put in the `on’ position simultaneously).
Next, suppose that after his experimentation, the physicist discovers that the following relation holds:
(Remember the Pythogorean theorem? It’s useful here!) Suppose he finds the above equation holds no matter what the specific values of and may be (i.e. whatever be the distances of the two arbitrarily selected points from the same origin).
In this case, the physicist declares that this universe is a -dimensional vector space, with respect to these and properties taken as the bases.
Why? Why does he call it a -dimensional universe? Why doesn’t he continue calling it a -dimensional universe?
Because, he can take a -dimensional graph paper by way of an abstract representation of how the quantities of the properties (or attributes) vary, plot these quantities and along the two Cartesian axes, and then use them to determine the third quantity from them. (In fact, he can use any two of these strengths to find out the third one.)
In particular, he happily and blithely ignores the fact that the object of which are mere properties (or attributes), actually is spread (or extended) over only a single dimension, viz., the -axis.
He still insists on calling this universe a -dimensional universe.
That’s all there is to this -dimensional nonsense. Really.
But what about the -dimensional space, you ask?
Well, the physicist just regards the extension and the position themselves to form the set of the physical properties under discussion! The physicist regards distance as a property, even if he is going to measure the strengths or magnitudes of the properties (i.e. distances, really speaking) only in reference to (i.e. positions)!!
But doesn’t that involve at least one kind of a circularity, you ask?
The answer is embedded right in the question.
Understand this part, and the entire mystification of physics based on the “multi-dimensional” whatever vaporizes away.
But don’t rely on the popular science paperbacks to tell you this simple truth, though!
Hopefully, the description above is not too dumbed down, and further, hopefully, it doesn’t have too significant an error. (It would be easy for me (or for that matter any one else) to commit an error—even a conceptual error—on this topic. So, if you spot something, please do point it out to me, and I will correct the description accordingly. On my part, I will come back sometime next week, and read this post afresh, and then decide whether what I wrote makes sense or not.)
A Song I Like:
For this time round, I am going to list a song even if I don’t actually evaluate it to be a very great song.
In fact, in violation of the time-honored traditions of this blog, what I am going to do is to list the video of a song. It’s the video of a 25+ years old song that I found I liked, when I checked it out recently. As to the song, well, it has only a nostalgia value to me. In fact, even the video, for the most part, has only a nostalgia value to me. The song is this:
(Hindi) “may se naa minaa se na saaki se…”
Music: Rajesh Roshan
Singers: Sadhana Sargam and Mohammad Aziz
Lyrics: Farooq Qaisar
Well, those were the technical details (regarding this song). To really quickly locate the song (and the video), forget the lyrics mentioned above. Instead, just google “aap ke aa jaane se,” and hit the first video link that the search throws up. (Yes, it’s the same song.)
As I said, I like this video mainly for its nostalgic value (to me). It instantaneously takes me back to the 1987–88 times. The other reasons are: the utter natural ease with which both the actors perform the dance here (esp. Neelam!). They both in fact look like they are authentically enjoying their dancing. Watch Neelam’s steps, in particular. She was reputed to be a good dancer, and you might think that this song must have been a cake-walk for her. Well, check out her thin (canvas-like) shoes, and the kind of rough ground in the mountains and in the fields over which she seems so effortlessly to take those steps. Govinda, in comparison, must have had it a bit easier (with his thicker, leather shoes), but in any case, in actuality, it must have been some pretty good hard work for both of them—it’s just that the hard work doesn’t show in the song. … Further, I also like the relative simplicity of the picturization. And, the catchy rhythm. Also, the absence, here, of those gaudy gestures which by now are so routine in Hindi film songs (and in fact were there even in the times of this song, and in fact also for about a decade or more earlier). I mean: those pelvic thrusts, that passing off of a thousand of people doing their PT exercises on a new, sprawling suburban street in Mumbai/Gurgaon/Lutyens’ Delhi as an instance of dance, etc.
… I don’t know if you end up liking this song or not. To me, however, it unmistakably takes me to the times when I was a freshly minted MTech from IIT Madras, was doing some good (also satisfying) work in NDT, had just recently bought a bike (the Yamaha RX 100), and was looking forward to life in general with far more enthusiasm (and in retrospect, even naivete) than I can manage to even fake these days. So, there.
As I said, drop a line if there are mistakes in the main post. Main mistake (or omission) corrected. As I said, drop a line if there are further mistakes in the main post. And, excuse me for some time, esp. the next week-end, esp. the next Saturday late night (IST). I may not find any time the next Sunday, because I would once again be in the middle of teaching a couple of new units over the next 2–3 weeks.]