Complex Analysis
Exploring Hidden Structure in Higher Dimensions
Fractal
A Discussion of Fractals
Fibonacci
Understanding the ubiquitous patterns of the universe
Fractal
Searching For The Mandelbrot In Higher Dimensions
Fractal
From Gaston Julia to Benoit Mandelbrot
Imagine a square of paper lying flat on your desk. I ask you to close your eyes. You hear the paper shift. When you open your eyes the paper doesn’t appear to have changed. What could I have done to it while you weren’t looking?
It’s obvious that I haven’t rotated the paper by 30 degrees, because then the paper would look different.
I also did not flip it over across a line connecting, say, one of the corners to the midpoint of another edge. The paper would look different if I had.
What I could have done, however, was rotate the paper clockwise or counterclockwise by any multiple of 90 degrees, or flipped it across either of the diagonal lines or the horizontal and vertical lines.
Flipping across any dashed line will not change the square.
A helpful way to visualize the transformations is to mark the corners of the square.
The last option is that to do nothing. This is called the identity transformation. Together, these are all called the symmetry transformations of the square.
I can combine symmetry transformations to make other symmetry transformations. For example, two flips across the line segment BD produces the identity, as do four successive 90 degree counterclockwise rotations. A flip about the vertical line followed by a flip about the horizontal line has the safe effect as a 180 degree rotation. In general, any combination of symmetry transformations will produce a symmetry transformation. The following table gives the rules for composing symmetry transformations:
We use “e” for the identity transformation.
In this table, R with subscripts 90, 180, and 270 denote counterclockwise rotations by 90, 180, and 270 degrees, H means a flip about the horizontal line, V is a flip about the vertical line, MD is a flip about the diagonal from the top left to the bottom right, and OD means a flip over the other diagonal. To find the product of A and B, go to the row of A and then over to the column of B. For example, H∘MD=R₉₀.
There are a few things you may notice by looking at the table:
We therefore say that the collection of symmetry transformations of a square, combined with composition, forms a mathematical structure called a group. This group is called D₄, the dihedral group for the square. These structures are the subject of this article.
A group ⟨G,*⟩ is a set G with a rule * for combining any two elements in G that satisfies the group axioms:
In the abstract we often suppress * and write a*b as ab and refer to * as multiplication.
An example of a group from everyday life is the set of “moves” that can be made on a Rubik’s cube under composition. Source.
It is not necessary for * to be commutative, meaning that a*b=b*a. You can see this by looking at the table of D₄, where H∘MD=R₉₀ but MD∘H=R₂₇₀. Groups where * is commutative are called abelian groups after Neils Abel.
Abelian groups are the exception rather than the rule. Another example of a non-abelian group is the symmetry transformations of a cube. Consider just rotations about the axes:
Source: Chegg
If I first rotate 90 degrees counterclockwise about the y-axis and then 90 degrees counterclockwise about the z-axis then his will have a different result than if I were to rotate 90 degrees about the z-axis and then 90 degrees about the y-axis.
Top row: Rotation 90 degrees about y followed by 90 degrees about z. Bottom row: 90 degree rotation about z followed by 90 degree rotation about y.
It is possible for an element to be its own inverse. Consider the group which consists of 0 and 1 with the operation of binary addition. Its table is:
Clearly 1 is its own inverse. This is also an abelian group. Don’t worry, most groups aren’t this boring.
Some more examples of groups include:
The 5th roots of unity, which solve x⁵-1=0
Here are some examples that are not groups:
A group is a lot more than just a blob that satisfies the four axioms. A group can have internal structure, and this structure can be very intricate. One of the basic problems in abstract algebra is to determine what the internal structure of a group looks like, since in the real world the groups that are actually studied are much larger and more complicated than the simple examples we’ve given here.
One of the basic types of internal structure is a subgroup. A group G has a subgroup H when H is a subset of G and:
If H≠G then H is said to be a proper subgroup. The subgroup of G consisting only of the identity is called the trivial subgroup.
A group of n elements where every element is obtained by raising one element to an integer power, {e, a, a², …, aⁿ⁻¹}, where e=a⁰=aⁿ, is called a cyclic group of order n generated by a. Consider the cyclic subgroup of order 6, {e,a,a²,a³,a⁴,a⁵}. Its proper subgroups are {e,a³} and {e,a²,a⁴}.
A non-abelian group may have commutative subgroups. Consider the square dihedral group that we discussed in the introduction. This group is not abelian but the subgroup of rotations is abelian and cyclic:
We now give two examples of group structure.
Even if a group G is not abelian, it may still be the case that there is a collection of elements of G that commute with everything in G. This collection is called the center of G. The center C is a subgroup of G. Proof:
Now suppose that f is a function whose domain and range are both G. A period of fis an element a∈G such that f(x)=f(ax) for all x∈G. The set P of periods of f is a subgroup of G. Proof:
We saw that cyclic groups are generated by a single element. When it’s possible to write every element of a group G as products of a (not necessarily proper) subset A of G then we say that A generates G and write this as G=⟨A⟩. The most “well, duh” proof you’ll ever see is the proof that all finite groups are generated by a finite generating set:
Every finite group is trivially generated by itself but it may also be generated by a proper subset. For example, the group G={e, a, b, b², ab, ab²} with the constraints a²=e, b³=e, ba=ab² is is generated by a and b so G=⟨{a,b}⟩.
We will finish off this article with an application.
The simplest way to transmit digital information is to encode it into binary strings of fixed length. No communication scheme is completely free from interference so there is always a possibility that the wrong data will be received. The method of maximum-likelihood decoding is a simple and effective approach to detecting and correcting transmission errors.
Let 𝔹ⁿ be the set of binary strings, or words, of length n. It is straightforward to show that 𝔹ⁿ is an abelian group under binary addition without carrying (so that for example 010+011=001). Note that every element is its own inverse. A code C is a subset of 𝔹ⁿ. The following is an example of a code in 𝔹⁶:
C = {000000, 001011, 010110, 011101, 100101, 101110, 110011, 111000}
The elements of a code are called codewords. Only codewords will be transmitted. An error is said to occur when interference changes a bit in a transmitted word. The distance d(a,b) between two codewords a and b is the number of digits in which two codewords differ. The minimum distance of a code is the smallest distance between any two of its codewords. For the example code just above, the minimum distance is three.
In the method of maximum-likelihood decoding, if we receive a word x, which may contain errors, the receiver should interpret x as the codeword a such that d(a,x) is a minimum. I will show that for a code of minimum distance m this can always (1) detect errors that change fewer than m bits and (2) correct errors that change ½(m-1) or fewer bits.
The American theoretical physicist Richard Feynman is one of the best-known scientists in the world (see this link). Feynman made several contributions to theoretical physics. These include: a new formulation of quantum mechanics using path integrals, the theory of quantum electrodynamics (including the development of his famous pictorial representations known as Feynman diagrams), his contributions to the explanation of the phenomenon of superfluidity of supercooled liquid helium, and his pioneering work in quantum computing and nanotechnology. In 1965, he was one of the recipients of the Nobel Prize in Physics.
Figure 1: The American theoretical physicist and Nobelist Richard Feynman (source).
In his biography about Feynman, the American-Canadian theoretical physicist Lawrence Krauss attributes the following quote to him:
“There is pleasure in recognizing old things from a new viewpoint.”
— Richard Feynman
In this article, based on Sakurai, I will introduce Feynman’s brilliant new approach to quantum theory, based on path integrals.
The first successful attempt to replicate the quantization of the atomic spectra was called matrix mechanics. It was developed in 1925 by the German theoretical physicist Werner Heisenberg. In the same year, the Austrian-Irish physicist Erwin Schrödinger created his wave mechanics. After a year, Schrödinger himself proved the equivalence of both approaches. However, the modern view of quantum mechanics, where states are mathematical entities in Hilbert space, is attributed mainly to the English theoretical physicist Paul Dirac. Driven by the development of quantum field theory, other formulations of quantum mechanics were developed.
Figure 2: Werner Heisenberg, Erwin Schrödinger and Paul Dirac.
One of these approaches was the path integral formulation, which is a generalization of the action principle of classical mechanics. In this formalism, the quantum transition between two states is obtained by “summing” (more precisely evaluating a functional integral) over all possible classical trajectories. This formulation, therefore, replaces the classical notion that a single spacetime trajectory is associated with the motion of a system. Though the development of the path integral formulation involved other scientists (such as Norbert Wiener and Paul Dirac), the complete method was provided by Feynman in 1948.
Figure 3: The article where Feynman introduced his new approach (source).
The abstract of the article “Space-Time Approach to Non-Relativistic Quantum Mechanics”, where Feynman introduced his new approach reads:
“[Q]uantum mechanics is formulated here in a different way. It is, however, mathematically equivalent to the familiar formulation. In quantum mechanics the probability of an event which can happen in several different ways is the absolute square of a sum of complex contributions, one from each alternative way. The probability that a particle will be found to have a path x(t) lying somewhere within a region of spacetime is the square of a sum of contributions, one from each path in the region. The contribution from a single path is postulated to be an exponential whose (imaginary) phase is the classical action (in units of h/2π) for the path in question. The total contribution from all paths reaching x, t from the past is the wave function ψ(x, t). This is shown to satisfy Schrödinger’s equation…”
To start, I will first introduce the concept of a quantum propagator.
Propagators in quantum mechanics and quantum field theory are functions that determine the probability amplitude for a particle to move from one position to another in a given interval of time (transition amplitude). The modulus squared of the transition amplitude is the transition probability.
Let the time-dependent X(t) be the position operator in the Heisenberg picture. Using the elegant (bra and ket) notation developed by Dirac, consider the following initial and final vector states
Equation 1: Transition between states.
which are eigenstates of the operator X(t):
The propagator corresponding to the transition in Eq. 1 is given by the following expression:
Equation 2: Propagator corresponding to the transition in Eq. 1.
This function (the propagator) is a transition amplitude.
Naturally, the probability that we get any result when we measure the position of a system (at some fixed time) must be equal to 1. A consequence of this is that position kets form a complete set:
Equation 3: The position kets at a fixed time form a complete set.
Note that the unit on the right-hand side is the identity operator. This relation can be used to write the propagator as a succession of integrals. In other words, we can write a transition as a combination of intermediary transitions. The simplest case is:
Equation 4: The use of Eq. 3 to re-express the propagator on the left-hand side as an integral over x’’. The integral represents a succession of two transitions for all possible intermediary positions x’’.
Using this compositional property of the propagator we may divide the time interval corresponding to a transition between two states
Equation 5: Transition between an initial and a final state.
into equal parts with the length
Equation 6: The time interval corresponding to a complete transition can be divided into equal parts with the length Δt.
and then write the transition amplitude as a composition of amplitudes:
Equation 7: Transition amplitude as a composition of amplitudes.
We are summing over all possible paths, with fixed endpoints. This is illustrated in the middle plot in Fig. 4.
Figure 4: Classical trajectory, discrete quantum trajectories, and continuous quantum trajectories (based on source).
The leftmost plot in Fig. 4 shows the classical mechanics’ unique path. To obtain this path consider a particle with an associated Lagrangian L(x, dx/dt). The Lagrangian equation of motion, which describes the path, is obtained by extremizing the action:
Equation 8: The classical path is obtained by extremizing this action.
This is called the Hamilton principle.
As we just saw, while in classical physics a particle motion has a definite path, in quantum mechanics all possible paths must be taken into account. The question is: in the limit h→ 0 how can we show that quantum mechanics reproduces classical mechanics?
In a 1933 paper, Paul Dirac made a mysterious remark which puzzled Feynman. Adapting the notation (following Sakurai), the statement was:
Equation 9: Dirac’s mysterious remark.
Feynman was troubled with the meaning of Dirac’s remark. What did he mean by “corresponds to”? This questioning led Feynman to the formulation of his powerful space-time approach to quantum mechanics.
Figure 5: Dirac’s mysterious remark in his 1933 paper “The Lagrangian in Quantum Mechanics” (source).
Following Sakurai, the action corresponding to a small segment of the path reads:
Equation 10: Action corresponding to a small segment of the path.
For a given path, we obtain the corresponding amplitude by multiplying such expressions. The propagator for the transition in Eq. 5 is then obtained by a sum over all paths:
Equation 11: The propagator for the transition in Eq. 5 is obtained by a sum over all possible paths. 
Figure 6: A few paths contributing to the propagator according to Eq. 11 (source).
For small h, the only relevant term of this sum corresponds to a path that does not vary if we deform it slightly (since h is small, the contribution of the other paths cancel out due to the strong oscillation of the exponentials). This means that this path satisfies Eq. 8:
Equation 12: The amplitude depends only on the classical path.
Therefore, according to Hamilton’s principle, the amplitude depends only on the classical path (more precisely, on a narrow strip that contains the classical path). Hence classical mechanics is recovered for small h, as it should!
Feynman rewrote Dirac’s Eq. 11 as a proportionality:
Equation 13: Feynman wrote Dirac’s assumption as a proportionality.
In his biography about Feynman, James Gleick describes the following conversation between the two great men:
“Feynman looked out the window and saw Dirac… He had a question that he had wanted to ask Dirac since before the war. He wandered out and sat down. A remark in a 1933 paper of Dirac’s had given Feynman a crucial clue toward his discovery of a quantum-mechanical version of the action in classical mechanics [see Eq. 9 above]. Dirac had written, but neither he nor anyone else had pursued this clue until Feynman discovered that the “[corresponds]” was, in fact, exactly proportional. There was a rigorous and potentially useful bond. Now, he asked Dirac whether the great man had known all along that the two quantities were proportional. “Are they?” Dirac said. Feynman said yes, they were. After a silence Dirac walked away.”
Figure 7: Richard Feynman talking with Paul Dirac at Relativity Conference in Warsaw (source).
Setting V=0 for simplicity, using a linear approximation for S(n, n-1) and integrating both sides of Eq. 13, one quickly obtains the prefactor 1/w(Δt):
Equation 14: Eq. 13 after the prefactor is evaluated.
The transition amplitude for the full path is obtained by integrating over all intermediate steps:
Equation 15: Transition amplitude Eq. 5 in Feynman’s formulation.
where:
Equation 16: Feynman’s path integral.
This is the so-called Feynman’s path integral and is the sum (or integral) of all possible paths. The corresponding diagram is shown in Fig. 4 (third plot) and Fig. 6.
Note here that in contrast to the other formulations of quantum mechanics, the probability amplitude is not a linear complex superposition of quantum states. It is a quantum superposition of entire alternative space-time histories (between fixed points). The idea here is that quantum mechanically there is a complex superposition of “alternative realities” (alternative histories). However, the exponential weighting, usually called amplitude,
is, in fact, an “amplitude density” since there is a continuous infinity of classical alternatives and we must therefore integrate over the space of classical paths. Since we have infinite classical paths, this space is infinite-dimensional and the integral is known to have pathological behavior.
Also, it should be noted that the trajectories do not need to obey the rules of special relativity: any trajectory is possible provided the endpoints are fixed (see figure below). Fig. 8 below illustrates the case of a single particle.
Figure 8: All paths are allowed in the path integral (source).
We shall now prove that Feynman’s approach is equivalent to Schrödinger wave mechanics. We start by writing:
Equation 17: Total amplitude expressed as a composition of two steps, one of them with infinitesimal time difference.
where we assumed:
Equation 18: Definition of ξ → 0.
We now rename the final position and final time as x and t + Δt, use ξ in Eq. 18, and expand the exponential and the amplitude (inside the integral) in powers of ξ. Collecting terms and performing trivial Gaussian integrals we obtain:
Equation 19: Schrödinger equation for Feynman’s propagator.
We conclude that Feynman’s propagator and Schrödinger’s propagator (obtained using his wave mechanics) are the same objects!
Thanks for reading and see you soon! As always, constructive criticism and feedback are always welcome!
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