POSSIBLE TYPES OF STATISTICAL INFORMATION ABOUT A SYSTEM 2
In modern science there is a tendency of generalization or extension of one principle to others. For example; the Schrödinger equation in the so-called one dimension (actually it contains a second order term; hence cannot be an equation in one dimension) is generalized (?) to three dimensions by adding two more terms for y and z dimensions (mathematically and physically it is a wrong procedure). We have discussed it in latter pages. While position and momentum are specific quantities, the generalizations are done by replacing these quantities with A and B. When a particular statement is changed to a general statement by following algebraic principles, the relationship between the quantities of the particular statement is not changed. However, physicists often bypass or over-look this mathematical rule. A and B could be any set of two quantities. Since they are not specified, it is easy to use them in any way one wants. Even if the two quantities are commutable, since they are not precisely described, it gives one the freedom to manipulate by claiming that they are not commutable and vice-versa. Modern science is full of such manipulations.
Here we give another example to prove that physics and modern mathematics are not always compatible. Bell’s Inequality is one of the important equations used by all quantum physicists. We will discuss it repeatedly for different purposes. Briefly the theory holds that if a system consists of an ensemble of particles having three Boolean properties A, B and C, and there is a reciprocal relationship between the values of measurement of A on two particles, the same type relationship exists between the particles with respect to the quantity B, the value of one particle measured and found to be a, and the value of another particle measured and found to be b, then the first particle must have started with state (A = a, B = b). In that event, the Theorem says that P (A, C) ≤ P (A, B) + P (B, C). In the case of classical particles, the theorem appears to be correct.
Quantum mechanically: P(A, C) = ½ sin2 (θ), where θ is the angle between the analyzers. Let an apparatus emit entangled photons that pass through separate polarization analysers. Let A, B and C be the events that a single photon will pass through analyzers with axis set at 00, 22.50, and 450 to vertical respectively. It can be proved that C → C.
Thus, according to Bell’s theorem: P(A, C) ≤ P(A, B) + P(B, C),
Or ½ sin2 (450) ≤ ½ sin2 (22.50) + ½ sin2 (22.50),
Or 0.25 ≤0.1464, which is clearly absurd.
This inequality has been used by quantum physicists to prove entanglement and distinguish quantum phenomena from classical phenomena. We will discuss it in detail to show that the above interpretation is wrong and the same set of mathematics is applicable to both macro and the micro world. The real reason for such deviation from common sense is that because of the nature of measurement, measuring one quantity affects the measurement of another. The order of measurement becomes important in such cases. Even in the macro world, the order of measurement leads to different results. However, the real implication of Bell’s original mathematics is much deeper and points to one underlying truth that will be discussed later.
A wave function is said to describe all possible states in which a particle may be found. To describe probability, some people give the example of a large, irregular thundercloud that fills up the sky. The darker the thundercloud, the greater the concentration of water vapor and dust at that point. Thus by simply looking at a thundercloud, we can rapidly estimate the probability of finding large concentrations of water and dust in certain parts of the sky. The thundercloud may be compared to a single electron’s wave function. Like a thundercloud, it fills up all space. Likewise, the greater its value at a point, the greater the probability of finding the electron there! Similarly, wave functions can be associated with large objects, like people. As one sits in his chair, he has a Schrödinger probability wave function. If we could somehow see his wave function, it would resemble a cloud very much in the shape of his body. However, some of the cloud would spread out all over space, out to Mars and even beyond the solar system, although it would be vanishingly small there. This means that there is a very large likelihood that his, in fact, sitting here in his chair and not on the planet Mars. Although part of his wave function has spread even beyond the Milky Way galaxy, there is only an infinitesimal chance that he is sitting in another galaxy. This description is highly misleading.
The mathematics for the above assumption is funny. Suppose we choose a fixed point A and walked in the north-eastern direction by 5 steps. We mark that point as B. There are an infinite number of ways of reaching the point B from A. For example, we can walk 4 steps to the north of A and then walk 3 steps to the east. We will reach at B. Similarly, we can walk 6 steps in the northern direction, 3 steps in the eastern direction and 2 steps in the Southern direction. We will reach at B. Alternatively; we can walk 8 steps in the northern direction, 6 steps in the eastern direction and 5 steps in the South-eastern direction. We will reach at B. It is presumed that since the vector addition or “superposition” of these paths, which are different sorts from the straight path, lead to the same point, the point B could be thought of as a superposition of paths of different sort from A. Since we are free to choose any of these paths, at any instant, we could be “here” or “there”. This description is highly misleading.
To put the above statement mathematically, we take a vector V which can be resolved into two vectors V1 and V2 along the directions 1 and 2, we can write: V = V1 + V2. If a unit of displacement along the direction 1 is represented by 1, then V1 = V11, wherein V1 denotes the magnitude of the displacement V1. Similarly, V2 = V22. Therefore:
V = V1 + V2 = V11 + V22. [1 and 2 are also denoted as (1,0) and (0,1) respectively].
This equation is also written as: V = λ1 + λ2, where λ is treated as the magnitude of the displacement. Here V is treated as a superposition of any standard vectors (1,0) and (0,1) with coefficients given by the numbers (ordered pair) (V1 , V2). This is the concept of a vector space. Here the vector has been represented in two dimensions. For three dimensions, this equation is written as V = λ1 + λ2 + λ3. For an n-tuple in n dimensions, the equation is written as V = λ1 + λ2 + λ3 +…… λn.
It is said that the choice of dimensions appropriate to a quantum mechanical problem depends on the number of independent possibilities the system possesses. In the case of polarization of light, there are only two possibilities. The same is true for electrons. But in the case of electrons, it is not dimensions, but spin. If we choose a direction and look at the electron’s spin in relation to that direction, then either its axis of rotation points along that direction or it is wholly in the reverse direction. Thus, electron spin is described as “up” and “down”. Scientists describe the spin of electron as something like that of a top, but different from it. In reality, it is something like the nodes of the Moon. At one node, Moon appear to be always going in the northern direction and at the other node, it always appears to be going in the southern direction. It is said that the value of “up” and “down” for an electron spin is always valid irrespective of the directions we may choose. There is no contradiction here, as direction is not important in the case of nodes. It is only the lay out of the two intersecting planes that is relevant. In many problems, the number of possibilities is said to be unbounded. Thus, scientists use infinite dimensional spaces to represent them. For this they use something called the Hilbert space. We will discuss about these later.
Any intelligent reader would have seen through the fallacy of the vector space. Still we are describing it again. Firstly, as we have described in the wave phenomena in later pages, superposition is a merger of two waves, which lose their own identity to create something different. What we see is the net effect, which is different from the individual effects. There are many ways in which it could occur at one point. But all waves do not stay in superposition. Similarly, the superposition is momentary, as the waves submit themselves to the local dynamics. Thus, only because there is a probability of two waves joining to cancel the effect of each other and merge to give a different picture, we cannot formulate a general principle such as the equation: V = λ1 + λ2 to cover all cases, because the resultant wave or flat surface is also transitory.
Secondly, the generalization of the equation V = λ1 + λ2 to V = λ1 + λ2 + λ3 +…… λn is mathematically wrong as explained below. Even though initially we mentioned 1 and 2 as directions, they are essentially dimensions, because they are perpendicular to each other. Direction is the information contained in the relative position of one point with respect to another point without the distance information. Directions may be either relative to some indicated reference (the violins in a full orchestra are typically seated to the left of the conductor), or absolute according to some previously agreed upon frame of reference (Kolkata lies due north-east of Puri). Direction is often indicated manually by an extended index finger or written as an arrow. On a vertically oriented sign representing a horizontal plane, such as a road sign, “forward” is usually indicated by an upward arrow. Mathematically, direction may be uniquely specified by a unit vector in a given basis, or equivalently by the angles made by the most direct path with respect to a specified set of axes. These angles can have any value and their inter-relationship can take an infinite number of values. But in the case of dimensions, they have to be at right angles to each other which remain invariant under mutual transformation.
According to Vishwakaema the perception that arises from length is the same that arises from the perception of breadth and height – thus they belong to the same class, so that the shape of the particle remains invariant under directional transformations. There is no fixed rule as to which of the three spreads constitutes either length or breadth or height. They are exchangeable in re-arrangement. Hence, they are treated as belonging to one class. These three directions have to be mutually perpendicular on the consideration of equilibrium of forces (for example, electric field and the corresponding magnetic field) and symmetry. Thus, these three directions are equated with “forward-backward”, “right-left”, and “up-down”, which remain invariant under mutual exchange of position. Thus, dimension is defined as the spread of an object in mutually perpendicular directions, which remains invariant under directional transformations. This definition leads to only three spatial dimensions with ten variants. For this reason, the general equation in three dimensions uses x, y, and z (and/or c) co-ordinates or at least third order terms (such as a3+3a2b+3ab2+b3), which implies that with regard to any frame of reference, they are not arbitrary directions, but fixed frames at right angles to one another, making them dimensions. A one dimensional geometric shape is impossible. A point has imperceptible dimension, but not zero dimensions. The modern definition of a one dimensional sphere or “one sphere” is not in conformity with this view. It cannot be exhibited physically, as anything other than a point or a straight line has a minimum of two dimensions.
While the mathematicians insist that a point has existence, but no dimensions, the Theoretical Physicists insist that the minimum perceptible dimension is the Planck length. Thus, they differ over the dimension of a point from the mathematicians. For a straight line, the modern mathematician uses the first order equation, ax + by + c = 0, which uses two co-ordinates, besides a constant. A second order equation always implies area in two dimensions. A three dimensional structure has volume, which can be expressed only by an equation of the third order. This is the reason why Born had to use the term “d3r” to describe the differential volume element in his equations.