Guide
The Calabi-Yau manifold described in the previous article, because of its special topological properties, becomes the core of an additional 6-dimensional compact space in explanatory string theory. However, the 6 Vikarabi-Chu manifold has a large number of topological forms, and the geometry is very complex and difficult to imagine intuitively.
So, in general, when focusing on interpreting physical images without having to delve into the mathematical properties of "extra 6-dimensional space", we use the simplest 6-dimensional torus in compaction instead of complex Calabi-Chu manifolds. In layman's terms, we replace unimaginably complex manifolds with relatively simple torus.
In this article, we start from the motion law of classical point particles to look at the rules of the game of "strings" in string theory.
introduction
The size of the nine-dimensional space is very different The law of opening and closing the string is different
Written by | Zhang Tianrong
Editor-in-charge | Ning Qian, Lu Haoran
<h1 class="pgc-h-arrow-right" data-track="19" >01 flat torus</h1>
When it comes to torus, we often think of the donut shape, but topologists prefer a torus that is depicted in a more abstract way. In Figure 1a, we draw the two-dimensional torus as a rectangle.
Figure 1: The process of forming a flat (2-dimensional) torus can be imagined as gluing the ends of a barrel-shaped torus into a "donut", but with substantial differences.
In the square of Figure a, the two edges corresponding to the "A" arrow will be glued together, and the two edges corresponding to the "B" arrow will also be glued together. That is, as seen in science fiction: when you walk out of the rectangle from the A side above the rectangle, you will appear on the A side below; when you pass through the B on the right side of the rectangle, you will appear on the B side on the left.
The resulting two-dimensional torus, known as abstract torus, or Flat Torus[1], is shaped differently from a donut. In fact, as shown in Figure 1: the two A-sides bond to form a cylinder, and there is nothing wrong with this first step. However, in the second step, when we glue the two ends of B of the column of Figure C together, it is impossible for us to get the shape of a real donut, that is, the smooth and wrinkle-free surface shown in Figure 1d. There's a deep reason for this: the abstract torus comes from a flat rectangle, which is flat in nature, with zero curvature, while the intrinsic curvature of a donut as you usually see is not zero. Abstract torus is different from the intrinsic nature of donuts.
Although an abstract torus cannot be smoothly embedded in three-dimensional space, it is easy to rely on imagination to understand its topological properties.
The simplest torus is a 1-dimensional circle, and Figure 1 constructs a two-dimensional torus, the method of which can be generalized to a higher dimension. For example, imagine a box with six faces: (A, A'), (B, B'), (C, C'), two parallel to each other. Imagine A and A' glued together, B and B', C and C' together, forming a 3-dimensional abstract torus. The same approach allows the construction of abstract toruses of any n-dimensionality.
<h1 class="pgc-h-arrow-right" data-track="124" >02 (classical) motion of strings in space-time</h1>
The Calabi-Yau manifold or abstract torus, as a 6-dimensional compact space, coupled with the familiar 4-dimensional space-time that unfolds on a large scale, constitutes the 10-dimensional space-time of string theory, which is the stage where "strings" are active. That is to say, there are two kinds of stages in string theory: a large stage in 4-dimensional space-time (that is, the real world in which we live) and a small stage in 6 dimensions. From a mathematical point of view, space in string theory is the Cartesian product of 4-dimensional space-time and 6-dimensional compact space.
In addition to the stage and the actors, there must be a script, that is, the rules of the game. For physics, the rules of the game are quantum and non-quantum (classical).
The laws of motion of strings in space can be generalized from the laws of motion of point particles. First, let's look at how to generalize the classical point particle rule to the classical string.
In Newtonian mechanics , the trajectory of a point particle is a line in 3-dimensional space that changes with time. In relativity, the trajectory of particles moving in 4-dimensional space-time is called the "world line", see Figure 2a. In string theory, a 0-dimensional point particle is replaced by a (smaller) 1-dimensional string motion. The trajectory of the string's motion in space-time is replaced by a trajectory plane, called the "world plane", as shown in 2b. Further, if the moving entity is two-dimensional (membrane), the trajectory of motion in space-time is called a "world body", as shown in Figure 2c.
Figure 2: Particle vs String (or Membrane)
The chord model has many advantages over the mathematical model of the sizeless point particle, one of which is to avoid the infinity problem of the point particle.
In classical electronics there are infinite difficulties. In classical physics , an electron can be treated as a globule with a radius of r , and the mass formula for an electron is m = e2/rc2. As r approaches 0, the mass becomes infinite. Finally, classical electron theory avoids this divergence by introducing the finite radius of electrons (non-point particles). In quantum field theory , the method of renormalization is required to eliminate infinity , but the gravitational field cannot be reformulated , so that it cannot be included in the Standard Model.
However, for string theory, renormalization becomes irrelevant, because strings are not points, strings have sizes, and naturally remove the problem of divergence of point particles.
< h1 class="pgc-h-arrow-right" data-track="125" >03 quantum chord-sum interaction</h1>
The analogy of the midpoint particle and string motion in Figure 2 is easy to generalize to other situations, including quantum strings and interactions. That is, the line at the point particle is replaced in string theory with a ribbon surface (open string) or a pipe face (closed string) (shown in Figure 2b).
For example, the world line in Figure 2a is the trajectory of a classical particle in 4-dimensional space-time. There is only one classical path between two fixed points, but if you consider quantum mechanics, there are infinitely many paths for a particle from A to B, and the classical path (the blue line) is just one of them (see Figure 3a). Figure 3b shows a similar situation in string theory: all possible world planes, except for the classical world plane represented by blue, contribute to the calculation of the quantum probability amplitude of strings A to B.
Figure 3: Path Integral (Quantization)
According to quantum field theory, particles in space-time are constantly annihilated and constantly produced. Interaction phenomena such as generation and annihilation are described and calculated using Feynman diagrams at all levels, and string theory is no exception, but as mentioned above, the corresponding line segments need to be replaced by "surfaces", see Figure 4.
Figure 4: String-to-string interaction in string theory and Feynman diagram
From a field theory perspective, another advantage of string theory is that it is more simplified compared to the Standard Model. Quantum field theory can mathematically have an infinite variety of particles and therefore correspond to an infinite variety of particles. For example, of the 61 elementary particles corresponding to the Standard Model, there are 61 different quantum field theories. In string theory, only a quantum field theory that describes "strings" is required, thus simplifying it conceptually.
<h1 class="pgc-h-arrow-right" data-track="126" >04 Extension of the membrane-chord concept</h1>
In Figure 2c, we have already mentioned the concept of "membrane", which first came from the supergravity theory associated with string theory. The membranes often talked about in today's string theory are p-films and D-films.
Physical entities called p-branes are produced by generalizing the concept of point particles to dimensions 1, 2, and beyond. For example, point particles can be thought of as 0-dimensional membranes, while strings can be thought of as 1-dimensional membranes, and "membranes" in the usual sense are 2-dimensional. In addition, there may be higher dimensional membranes.
P-membranes are dynamic objects that travel through space-time according to the rules of quantum mechanics. They carry mass and other properties, such as charge. A p-film travels through space-time and sweeps out the volume of the (p+1) dimension, called the worldvolume, as shown in Figure 2c.
Another type of membrane is called d-brane, which represents a membrane that meets the Dirichlet boundary condition. D-membranes are an important type of membrane in string theory, which is related to the motion of open strings in space-time. When the open string travels through space-time, the end of the open string must be on the D film. The study of D-membranes has yielded important results related to duality, which will be briefly introduced in the next article.
Figure 5: Open string on D membrane
<h1 class="pgc-h-arrow-right" data-track="127" >05 The particularity of the string's motion in compact space</h1>
As mentioned at the beginning of this article, the space of string theory is divided into a large space for stretching and a small space for curling. Using the analogy of the previous article "Ants on a cable line": the cable line we see is a large space of 1 dimension, while the ants on the line can see another dimension of curled circles (small spaces).
The large space-time in string theory is 4-dimensional, and the curled and compact small space is 6-dimensional. Neither the 4-dimensional nor the 6-dimensional can be shown in a flat image, but, to facilitate the interpretation of the concept, we reduce the 10-dimensional space-time of string theory to a long cylinder in Figure 6a that looks like a 2-dimensional cable line in the eyes of an ant. The dimension along the direction of the cable line represents the 4-dimensional large space-time; the cross-sectional circle of the electric wire represents the 6-dimensional small space. Using such a metaphor, the open and closed strings in 10-dimensional space-time are small ants on the cable line.
That is, the infinitely extended x-direction in Figure 6 represents the 4-dimensional space-time we know, and the small circle curled in the y-direction represents 6 additional small dimensions. This 6-dimensional space can be a Calabi-Chu manifold, or it can simply understand the 6-dimensional abstract torus described in the cost section. For a 6-dimensional torus, the R in the figure should be understood to represent 6 values instead of 1 value.
Figure 6: Schematic of string theory space and "orbitals"
From point particles to string theory, not all physical quantities have corresponding analogies, and the particularity of strings produces properties that are not found in some point particle models, and we take the "orbit number" [2] of closed strings in compact space as an example, which is a physical quantity that is not found in the standard model of 4-dimensional space-time.
As shown in Figure 6, 10-dimensional space-time is divided into a large and a small, where the motion of the string can also be discussed from the motion of these two aspects. That is to say: the string, in addition to moving in a large 4-dimensional space-time, can also move around the compact space. This movement of closed strings is particularly special, because closed strings may have a special state, that is, around a certain (or more) compact dimensions, you can wrap around 1, 2 or many (N) circles, see Figure 6b. Therefore, the closed string has an additional quantum state, which is characterized by a new quantum number, called the "winding number".
There is no concept of windings, because open strings are topologically equivalent to a point and cannot be "wound" up.
When closed strings wound in the compact dimension interact, the total number of wounds is a conservation quantity.
The figure above Figure 6b gives the example of the windings w=0, +2, +1, -1, and the following figure represents a process of change (equivalent) of the closed string (from left to right) that can be used to simply illustrate the conservation of the windings: at the beginning, the closed string is simply placed on the cylinder, there is no circle, so w = 0. Points A and B on the closed string interact closely and become two closed strings of w=1 and w=-1, but the sum of the windings is still 0.
How, do you understand the difference between point particles and string gameplay? Well understood isn't it? In the next article, we will introduce duality.
Resources:
[1] https://en.wikipedia.org/wiki/Torus
[2] https://en.wikipedia.org/wiki/Winding_number