Derivatives with respect to a Supernumber?

["pi"mp]

So I've been trying to think about some papers in Supersymmetry and I need to somehow define a derivative of a supernumber, with respect to another supernumber. I mean a supernumber to be a number with an ordinary "body" and a "soul" which is a product of an even number of Grassmann numbers. Call it $z=z_{B}+z_{S}$.

I know what properties to expect from a derivative with respect to ordinary numbers and Grassmann numbers *separately* but I'm lost on how to combine them. In addition, I would like the body and soul to be complex.
Thanks for any tips

 

[mathman]

"body" and "soul" for numbers?

 

["pi"mp]

haha indeed! Awesome names. A supernumber contains a "body" which is simply an ordinary complex number, and a "soul" which is a sum of even products of Grassmann numbers.

 

[bahamagreen]

Does this help?
Chapter 3, section 3.2.2 page 23... here?

 

["pi"mp]

That's almost exactly what I want. Except I know how to differentiate with respect to bosonic and fermionic variables separately; I'm just not sure how to define a derivative with respect to a number that is both bosonic and fermionic.

 

["pi"mp]

My only idea would be to try to prove something analogous to the Cauchy-Riemann equations but I haven't had any luck. In other words, perhaps I could differentiate the body and soul of a function with respect to bodies and souls separately and then find some relation between them. Has anyone heard of anything along these lines?

 

[mathman]

I am not at all familiar with these concepts. I suggest that you go back to the basic definition of derivative as the limit of difference quotients, in particular define a difference quotient.

 

[Demystifier]

My only idea would be to try to prove something analogous to the Cauchy-Riemann equations but I haven't had any luck.

What turned out to be a problem?

 

["pi"mp]

Probably the one where I'm just not clever enough :) In complex analysis, you consider a function $f(z)=u(x,y)+i v(x,y)$ and the Cauchy-Riemann Equations give relations on real valued functions u and v. I want to think of a supernumber's body and soul as analogous to x and y, respectively above. Likewise, I'd like to take a function of that supernumber and think of its ordinary and fermionic parts as analogous to u and v from above.

Ultimately, I want something analogous to

$$\frac{\partial}{\partial z}=\frac{1}{2}\bigg(\frac{\partial}{\partial x}-i \frac{\partial}{\partial y}\bigg)$$

which would tell me exactly how to differentiate with respect to a supernumber in terms of the derivatives wrt the ordinary and fermionic parts, which I know how to do. However, the imaginary part of a holomorphic function is still a real-valued function whereas in my case, the "soul" of a superfunction may depend wildly on a bunch of Grassmann numbers.

 

[samalkhaiat]

My only idea would be to try to prove something analogous to the Cauchy-Riemann equations but I haven't had any luck. In other words, perhaps I could differentiate the body and soul of a function with respect to bodies and souls separately and then find some relation between them. Has anyone heard of anything along these lines?

All these stuff are discussed nicely in the first chapter of the classic textbook “Supermanifolds” by Bryce DeWitt, Cam. Univ. Press (1992). And, by the way super-analytic functions have no body-soul decomposition, rather they admit even(commuting)-odd(anticommuting) decomposition.

 

["pi"mp]

Right, I should be saying even vs. odd when it comes to super-analytic functions. However, from what I can see, DeWitt deals only with differentiation w.r.t even or odd variables separately but doesn't hint at how to combine them. Am I missing something? Have you seen this in there?

 

[Demystifier]

classic textbook “Supermanifolds” by Bryce DeWitt, Cam. Univ. Press (1992).

Another book on the same subject is
F.A. Berezin, Introduction to Superanalysis (1987)

 

["pi"mp]

After all the help on this page, I think I've finally got this straightened out. Like someone pointed out, my main downfall seemed to be trying to think about super-analytic functions as having "bodies" and "souls" when in fact, only an even/odd decomposition is possible. That helps a lot. Also, maybe it's just me but I found Berezin a lot easier to digest than DeWitt. Thanks a lot everyone!

 

[samalkhaiat]

After all the help on this page, I think I've finally got this straightened out. Like someone pointed out, my main downfall seemed to be trying to think about super-analytic functions as having "bodies" and "souls" when in fact, only an even/odd decomposition is possible. That helps a lot. Also, maybe it's just me but I found Berezin a lot easier to digest than DeWitt. Thanks a lot everyone!

Now, I am not sure how much you know about the subject! The whole exercise is to parametrize the real superspace $\mathbb{ R }^{ p | q } = \mathbb{ R }^{ p }_{ c } \times \mathbb{ R }^{ q }_{ a }$ by $p$ c-number coordinates $x^{ n }$ and $q$ a-number coordinates $\theta^{ \alpha }$. A supernumber-valued function $f : \mathbb{ R }^{ p | q } \to \Lambda_{ \infty }$ is called super-analytic if the following Taylor series converges: $$f ( z ) \equiv f ( x , \theta ) = \sum_{ k = 0 }^{ \infty } f_{ A_{1} A_{ 2 } \cdots A_{ k } } \ z^{ A_{ 1 } } z^{ A_{ 2 } } \cdots z^{ A_{ k } } , \ \ f_{ A_{ 1 } \cdots A_{ k } } \in \Lambda_{ \infty } .$$ With a bit of work, you can rewrite the series as $$f ( x , \theta ) = f_{ 0 } ( x ) + \sum_{ k = 1 }^{ q } \frac{ 1 }{ k ! } f_{ [ \alpha_{ 1 } \cdots \alpha_{ k } ] } ( x ) \ \theta^{ \alpha_{ 1 } } \theta^{ \alpha_{ 2 } } \cdots \theta^{ \alpha_{ k } } ,$$ with all the $f$’s being superfunctions of $\mathbb{ R }^{ p }_{ c }$. Now, if you want to work with the first expansion, be my guest and use the following properties $$\partial_{ B } z^{ A } = \delta_{ B }{}^{ A } , \ \ \ \partial_{ A } \partial_{ B } = ( - 1 )^{ \epsilon ( A ) \ \epsilon ( B ) } \partial_{ B } \partial_{ A } ,$$ where $\epsilon ( A ) \equiv \epsilon ( z^{ A } )$ is Grassmann parity of the coordinates: $\epsilon ( \alpha ) = 1, \ \ \epsilon ( n ) = 0$. And for superfunctions $f$ and $g$, you have $$\partial_{ A } ( f \ g ) = ( \partial_{ A } f ) \ g + ( - 1 )^{ \epsilon ( A ) \ \epsilon ( f ) } f \ ( \partial_{ A } g ) ,$$ where $\epsilon ( f_{ \mbox{ even } } ) = 0$, $\epsilon ( f_{ \mbox{ odd } } ) = 1$ and $\epsilon ( \partial_{ A } f ) = \epsilon ( A ) + \epsilon ( f ) \ \ \mbox{ mod } \ 2$. You can also show that the action of complex conjugation on derivatives is given by $$( \partial_{ A } f )^{ * } = ( - 1 )^{ \epsilon ( A ) \left( 1 + \epsilon ( f ) \right) } \ \partial_{ A } f^{ * } .$$

Sam

 

["pi"mp]

In your first line, should your z's be the $\theta^{\alpha_{i}}$? If not, I don't understand where they come from. Or are the z's general supernumbers?

 

[samalkhaiat]

In your first line, should your z's be the $\theta^{\alpha_{i}}$? If not, I don't understand where they come from. Or are the z's general supernumbers?

So, as I expected, you don't know even the elementary concepts involved in here. What does the equation $\mathbb{ R }^{ p | q } = \mathbb{ R }^{ p }_{ c } \times \mathbb{ R }^{ q }_{ a }$ mean? As I said before, the real superspace $\mathbb{ R }^{ p | q }$ is parametrized by the coordinates $z^{ A } = ( x^{ n } , \theta^{ \alpha } )$, where $x^{ n } \in \mathbb{ R }^{ p }_{ c } , \ \ n =1 , 2 , \cdots , p$ are the bosonic (commuting) coordinates [they are called c-numbers], and $\theta^{ \alpha } \in \mathbb{ R }^{ q }_{ a } , \ \ \alpha = 1 , 2 , \cdots , q$ are the fermionic (anticommuting) coordinates [these are called a-numbers].

 

["pi"mp]

Well, you're right; I certainly don't understand this subject like I need to. I was really just referring to earlier, when I was trying to talk about differentiating a super-analytic function by a body and a soul. So let me see if I'm getting this: the coordinates on real superspace are $(x^{n}, \theta^{\alpha})$ with the first being bosonic and the last being bosonic. A general super-analytic function F is a linear combination of all possible products of the $\theta^{\alpha}$ where the coefficients are superfunctions of just the x's. The indices on the superfunctions must anti-commute since the $\theta^{\alpha}$ do. Then, to take a derivative with respect to the x's, you simply take ordinary derivatives of these coefficient superfunctions within the expression. To take a derivative with respect to the $\theta^{\alpha}$, you use the usual right/left derivatives discussed in Berezin and DeWitt. Am I starting to get on the right track?

Really, all I need to do is compute Jacobians using superdeterminants, which of course requires a matrix of derivatives. I think Berezin hopefully helped me understand how to take these but I would like to understand the basics. So I appreciate your comments.

 

[samalkhaiat]

Well, you're right; I certainly don't understand this subject like I need to. I was really just referring to earlier, when I was trying to talk about differentiating a super-analytic function by a body and a soul. So let me see if I'm getting this: the coordinates on real superspace are $(x^{n}, \theta^{\alpha})$ with the first being bosonic and the last being bosonic. A general super-analytic function F is a linear combination of all possible products of the $\theta^{\alpha}$ where the coefficients are superfunctions of just the x's. The indices on the superfunctions must anti-commute since the $\theta^{\alpha}$ do. Then, to take a derivative with respect to the x's, you simply take ordinary derivatives of these coefficient superfunctions within the expression. To take a derivative with respect to the $\theta^{\alpha}$, you use the usual right/left derivatives discussed in Berezin and DeWitt. Am I starting to get on the right track?

That is okay.

Really, all I need to do is compute Jacobians using superdeterminants, which of course requires a matrix of derivatives. I think Berezin hopefully helped me understand how to take these but I would like to understand the basics. So I appreciate your comments.

Supermatrices, superdeterminants(which also called Berezinian in modern works) and the operations on them are discussed in the above-mentioned two textbooks.

 

["pi"mp]

I'm curious, what happens if your coordinates parameterize *complex* superspace instead of real? How do the derivatives behave differently?

 

[samalkhaiat]

I'm curious, what happens if your coordinates parameterize *complex* superspace instead of real? How do the derivatives behave differently?

The rules of the derivatives do not change if the superfunctions are complex-valued functions.