Tensor #38. All Possible Operations텐서 #38. 가능한 모든 연산

These are my notes on XylyXylyX’s What is Tensor lectures. This post covers lecture 39 of the playlist, All Possible Operations. Up to the previous post we learned to read arrows, stacks, thumbtacks, sheaves, and honeycombs as pictures. This time we pick two of these objects and find every operation that can be made from them. We stay in three dimensions.

There are only two operations

On a 3-dimensional vector space $V$, the objects we have built so far are elements of the following spaces.

The one-dimensional spaces $\La{3}V^*$ and $\La{3}V$ are added to these, and we will see the sheaf shortly. Between these objects only two operations are available. One is the pairing of the dual space and the other is the wedge product. A pairing puts a vector into an empty slot of a form, so either a number comes out or a form of lower degree is left. A wedge product adds the degrees of objects on the same side. What the lecture does is go through the possible pairs one by one and organize what the result is and how to read it as a picture.

The sheaf is another face of the honeycomb

A sheaf is an arrow made into a density by multiplying in one 3-form basis $dx^1 \wedge dx^2 \wedge dx^3$. Only the directions of the lines and the number of lines per unit area carry meaning, and the length of a line carries none. So it must not be drawn like a vector field made of finite arrows. We must read it as a bundle of endlessly long parallel lines. This sheaf lives in the same space as the honeycomb. There are four correspondences that hold without a metric.

The first one is the correspondence between honeycomb and sheaf. The second says that a stack lives in the same space as a thumbtack made a density, the third says that an arrow is a honeycomb made a capacity, and the fourth says that a thumbtack is a stack made a capacity. Here a density means multiplying in the 3-form basis, and a capacity means multiplying in the 3-vector basis $\partial_1 \wedge \partial_2 \wedge \partial_3$. This is a different correspondence from the Hodge dual, which needs a metric. Writing the first one in components, from the sheaf $S^\alpha \, \partial_\alpha \otimes (dx^1 \wedge dx^2 \wedge dx^3)$ we put $\partial_\alpha$ into the 3-form and get the honeycomb $\tfrac{1}{2} \varepsilon_{\alpha\beta\gamma} S^\alpha \, dx^\beta \wedge dx^\gamma$. (This component formula is added to check the correspondence and was not covered in the lecture.)

Q. May I treat a sheaf and a honeycomb as the same object?
A. They are elements of the same space, and fixing one fixes the other. Written as a honeycomb it becomes a computation among forms, and written as a sheaf it can be read as a picture of arrows. We choose by turns the side that is easy to compute and the side that is easy to draw.

The two operations that make a scalar

The first is the pairing of an arrow and a stack. Putting the arrow $a^\mu \partial_\mu$ into the stack $b_\mu \, dx^\mu$ gives one number $b_\mu a^\mu$. As a picture, it is counting how many sheets of the stack the arrow pierces.

The second is the pairing of a sheaf and a thumbtack. We count how many lines of the sheaf pass through the area of the thumbtack, positive if the directions agree and negative if opposite. Written as a honeycomb, it is counting how many tubes of the honeycomb the thumbtack cuts. Both are counted values, so they do not change however we change the coordinates. A coordinate change has no way to move a tube from the inside to the outside.

The four operations made by wedge products

Objects on the same side take a wedge product.

The wedge of two arrows is the parallelogram piece the two arrows span, that is, a thumbtack. The wedge of two stacks is a honeycomb, the picture where the lines in which the sheets of the two stacks meet gather into tubes. This honeycomb can also be read as a sheaf.

The other two raise the degree to 3. The wedge of an arrow and a thumbtack is a 3-vector, and in three dimensions the 3-vectors form a one-dimensional space, so a single number $q$ is left as the coefficient.

Sweeping the thumbtack along the arrow makes one parallelepiped cell, and its volume is $q$. This $q$ is a number that multiplies the 3-vector basis, so it is a scalar capacity. On the other side, the wedge of a stack and a honeycomb is a 3-form, a scalar density. The sheets of the stack cut the lines of the sheaf into points, and the number of points per unit volume is its value.

The two operations made by inserting

There are two cases where the pairing leaves one object instead of making a number. Putting the arrow $a^\gamma \partial_\gamma$ into the honeycomb $S_{\alpha\beta} \, dx^\alpha \wedge dx^\beta$ leaves one empty slot and gives a stack.

Putting the stack $b_\alpha \, dx^\alpha$ into the thumbtack $T^{\alpha\beta} \partial_\alpha \wedge \partial_\beta$ gives an arrow.

If we use the sheaf in place of the honeycomb, the first becomes the operation that makes a stack from an arrow and a sheaf. Altogether that makes eight.

Cross product and dot product

Following Weinreich, the lecture names these operations. This approach sorts the objects into two types by shape. The arrow and the sheaf are arrow type, and the stack and the thumbtack are flat type. A stack looks like piled sheets and a thumbtack is a small flat piece, so both are grouped as flat type. A product of the same type is written as a cross product $\times$, and a product of different types as a dot product $\cdot$. The eight operations read like this.

• arrow $\cdot$ stack = scalar
• arrow $\times$ arrow = thumbtack
• stack $\times$ stack = sheaf
• arrow $\times$ sheaf = stack
• stack $\times$ thumbtack = arrow
• arrow $\cdot$ thumbtack = scalar capacity
• stack $\cdot$ sheaf = scalar density
• sheaf $\cdot$ thumbtack = scalar

The honeycomb does not enter this naming scheme. The honeycomb belongs to the MTW textbook side, and Weinreich draws the same-space sheaf instead.

Q. The wedge of two stacks is a honeycomb, so why is it written as a sheaf here?
A. Because they are elements of the same space. As the lecture puts it, speaking in this naming scheme needs one more step: rewriting the result of the wedge as a sheaf by the correspondence above. We can compute on the form side and convert to a sheaf only when drawing the result.

In the toy below, pick two objects. Each cell of the table shows the name, the degree count, and how to read it as a picture.

Q. Why is the product of two thumbtacks, or of two sheaves, 0?
A. In three dimensions the wedge of two 2-vectors would be a 4-vector, and there is no degree 4 in three dimensions, so it is 0. Two honeycombs are the same. There is also nothing to pair with. (This reason is added to fill the table and was not covered in the lecture.)

Making the products as pictures

The lecture builds the two products that are hard to draw.

The first is arrow $\times$ sheaf = stack. Place the arrow inside the sheaf and adjust the spacing so that two lines of the sheaf pass through the base and the tip of the arrow. Only the spacing changes with the density kept equal, so the sheaf stays the same. The plane made by the arrow and the direction of the lines becomes one sheet of the stack, and the neighboring sheets are placed to fit that spacing. The direction is fixed by the right-hand rule. This stack equals reading $c(x)$ as the number of lines of the sheaf passing through the parallelogram spanned by the arrow $a$ and $x$. (This reading of the formula puts the component formula $c_\beta = a^\alpha S_{\alpha\beta}$ above into words and was not covered in the lecture.)

The second is stack $\times$ thumbtack = arrow. Only the area of a thumbtack matters, so we may change its shape freely. Squeeze it into a rectangle that fits exactly between two neighboring sheets of the stack, and the edge lying in the sheet becomes the resulting arrow. The direction is fixed by the right-hand rule from the stack to the thumbtack, and reversing the order reverses it. In a formula, when the thumbtack has the form $u \wedge c$ with $b(u) = 1$ and $b(c) = 0$, the result $c^\beta = b_\alpha T^{\alpha\beta}$ is $c$. (This check is added and was not covered in the lecture.)

Scalars come in three kinds

The numbers made by the eight operations come in three kinds. The ordinary scalars made by the pairings of arrow with stack and of sheaf with thumbtack do not change when the coordinates change. The scalar capacity made by the wedge of an arrow and a thumbtack and the scalar density made by the wedge of a stack and a sheaf change their values when the coordinates change. But capacity and density change in opposite ways, so multiplying them gives an ordinary scalar.

The lecture says that an ordinary scalar can be seen as a capacity times a density. In the weights of post 31, the density is $+1$ and the capacity is $-1$, and multiplying them makes the weight 0. (Rewriting this with weights is a supplement to match the notation of the earlier post and was not covered in the lecture.)

Summary

What we checked in this post is the following.

• Only two operations are available between the objects, pairing and wedge, and picking two objects in three dimensions gives eight operations.
• The pairings that give a number are arrow with stack and sheaf with thumbtack, and these are counted values, so they are the same in any coordinates.
• Among wedge products, two arrows give a thumbtack, two stacks give a honeycomb (sheaf), an arrow and a thumbtack give a scalar capacity, and a stack and a sheaf give a scalar density.
• Among products made by inserting, an arrow and a sheaf give a stack, and a stack and a thumbtack give an arrow.
• In Weinreich’s naming, products of the same type are cross products and products of different types are dot products.

In the next post, lecture 40 closes the series, and we read the basis of each object off a small cube.