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Dirac Spinors, Grassmann Numbers and $SL(2,\mathbb{C})$ actions

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Recently, I've learned that the clifford algebra can be regarded as the quantization of the grassmann algebra from the following two papers of Berezin 

'Classical spin and Grassmann algebra'

http://www.jetpletters.ac.ru/ps/1476/article_22521.shtml

'Particle spin dynamics as the grassmann variant of classical mechanics'

http://www.sciencedirect.com/science/article/pii/0003491677903359

I've also noticed that when doing classical Dirac fields, sometimes they are treated as complex-valued spinors but sometimes they are treated as grassmann-valued spinors. They are treated as complex-valued spinors because they are the representation of the group $SL(2,\mathbb{C})$. But when we are dealing with the canonical and path-integral quantization of Dirac fields, we have to treat them as grassmann-valued spinors. 

Can anyone give me any mathematics oriented textbook or lecture notes explaining such a representation of $SL(2,\mathbb{C})$ over grassmann algebra? If there were such a representation, does it make any sense to construct grassmann valued spin up and spin down states in non-relativistic quantum mechanics? 

asked Feb 14, 2016 in Resources and References by XIaoyiJing (50 points) [ revision history ]
edited Feb 21, 2016 by XIaoyiJing

1 Answer

+ 3 like - 0 dislike

I realized that it would be time consuming to post the final answer with all those supermathematics details here. 

I would like to have a brief summary of the answer. 

Let us denote the ($\mathcal{N}=1$) super Poincare group by $S\Pi$, and denote the set of real (complex) even Grassmann numbers by $\mathbb{R}_{c}$ ($\mathbb{C}_{c}$) and the set of real (complex) odd Grassmann numbers by $\mathbb{R}_{a}$ ($\mathbb{C}_{c}$). We define the superspace $\mathbb{R}^{4|4}$ as $(\mathbb{R}_{c})^{4}\times (\mathbb{R}_{a})^{4}$.

In addition, we define the group $SL(2,\mathbb{C}_{c})$ as the special linear group with entries being $\mathbb{C}_{c}$ numbers. Similarly, the group $SO(3,1|\mathbb{R}_{c})^{+}$ is the Lorentz group with entries being commutative real Grassmann numbers.

It can be shown that $SL(2,\mathbb{C}_{c})/\mathbb{Z}_{2}=SO(3,1|\mathbb{R}_{c})^{+}$. The full super Poincare group is $S\Pi=T^{4|0}(\mathbb{R}_{c})\cup SO(3,1|\mathbb{R}_{c})^{+}\cup T^{0|4}(\mathbb{C}_{a})$. i.e. $S\Pi$ has three subgroups which are identified as translation on even coordinates, Lorentz transformation on even and odd coordinates, and translation on odd coordinates, respectively. 

It follows that we can prove that the superspace $\mathbb{R}^{4|4}$ is isomorphic the coset space $S\Pi/SO(3,1|\mathbb{R}_{c})^{+}$.

Remark: Since we are talking about $\mathcal{N}=1$ super Poincare group, we should use the identification $\mathbb{C}_{a}=\mathbb{R}_{a}\times\mathbb{R}_{a}$, and so $(\mathbb{R}_{c})^{4}\times(\mathbb{C}_{a})^{2}=\mathbb{R}^{4|4}$. In this way, it is possible to regard the odd coordinates as Majorana spinors, which will be clarified as follows.

An arbitrary group element of $S\Pi$ is of the form 

$g(b,\epsilon,\bar{\epsilon},K)=\exp\left[-i(-b^{a}P_{a}+\frac{1}{2}K^{ab}j_{ab}+\epsilon^{\alpha}q_{\alpha}+\bar{\epsilon}^{\dot{\alpha}}\bar{q}_{\dot{\alpha}})\right]$, 

where $\left\{P_{a},j_{ab},q_{\alpha},\bar{q}_{\dot{\alpha}}\right\}$ is a chosen basis of the Poincare superalgebra $\mathcal{G}$ over $\mathbb{C}$, the parameters $b^{a}$ and $K^{ab}$ are Grassmann even numbers, and $\epsilon_{\alpha}$ and $\bar{\epsilon}_{\dot{\alpha}}=(\epsilon_{\alpha})^{\ast}$ are Grassmann odd numbers.

The $\mathcal{N}=1$ Poincare superalgebra is a $\mathbb{Z}_{2}$-graded Lie algebra over $\mathbb{C}$, such that 

$[P_{a},P_{b}]=0$,    

$[j_{ab},P_{c}]=i\eta_{ac}P_{b}-i\eta_{bc}P_{a}$,

$[j_{ab},j_{cd}]=i\eta_{ac}j_{bd}-i\eta_{ad}j_{bc}+i\eta_{bd}j_{ac}-i\eta_{bc}j_{ad}$,

$[j_{ab},q_{\alpha}]=i(\sigma_{ab})^{\beta}_{\alpha}q_{\beta}$,

$[j_{ab},\bar{q}^{\dot{\alpha}}]=i(\bar{\sigma}_{ab})^{\dot{\alpha}}_{\dot{\beta}}\bar{q}^{\dot{\beta}}$,

$[P_{a},q_{\alpha}]=0$, $[P_{a},\bar{q}^{\dot{\alpha}}]=0$,

$\left\{q_{\alpha},q_{\beta}\right\}=0$,

$\left\{\bar{q}_{\dot{\alpha}},\bar{q}_{\dot{\beta}}\right\}=0$,

$\left\{q_{\alpha},\bar{q}_{\dot{\alpha}}\right\}=(\sigma^{a})_{\alpha\dot{\alpha}}P_{a}$.

This is simply a $\mathbb{Z}_{2}$-graded vector space over $\mathbb{C}$, denoted as $\mathcal{G}={}^{0}\mathcal{G}\oplus{}^{1}\mathcal{G}$, with a graded Lie bracket. The even generators $P_{a}$ and $j_{ab}$ form the ordinary Poincare algebra $\mathfrak{g}$ with ordinary Lie bracket $[\cdots,\cdots]$. The odd generators $q_{\alpha}$ and $\bar{q}_{\dot{\alpha}}$ are defined to anticommute with themselves and form a $\mathfrak{g}$-module via adjoint action.

While considering a generalization of ordinary Lie group to contain odd degrees of freedom, it is hopeless to use the exponential of the Poincare superalgebra since we do not have a generalization of Baker-Hausdorff formula for graded Lie brackets.

Instead, we may consider its Grassmann-shell, denoted as $\mathbf{G}$, which is defined as $\mathbf{G}=\Lambda_{\infty}\otimes\mathcal{G}$, where $\Lambda_{\infty}$ is the Grassmann algebra. This is a $\mathbb{Z}_{2}$-graded vector space over over $\Lambda_{\infty}$. i.e. $\mathbf{G}={}^{0}(\Lambda_{\infty}\otimes\mathcal{G})\oplus{}^{1}(\Lambda_{\infty}\otimes\mathcal{G})$. The first (second) summand is called the even (odd) part of the Grassmann-shell. In addition, we enforce a graded scalar multiplication rule such that any $\mathbb{C}_{a}$-valued Grassmann number anticommutes with odd vectors in the Grassmann-shell. 

It is easy to show that the even part of the Grassmann-shell is closed under ordinatry Lie bracket multiplication $[\cdots,\cdots]$. Using the even part of the Grassmann-shell, ${}^{0}(\Lambda_{\infty}\otimes\mathcal{G})$, with its ordinary (antisymmetric) commutator $[\cdots,\cdots]$, we can obtain a generalization of ordinary Lie group, which is called the super Lie group, whose local coordinates are labelled with both commutative numbers and anticommutative numbers. Elements in the super Lie group can be obtained via the exponential of elements in the even part of the Grassmann-shell ${}^{0}(\Lambda_{\infty}\otimes\mathcal{G})$.

In our case, the even numbers $b_{a}$ and $K_{ab}$ are not necessarily real numbers, but can be any Grassmann-even numbers in $\mathbb{R}_{c}$ and odd numbers $\epsilon^{\alpha}$ and $\bar{\epsilon}^{\dot{\alpha}}$ are from $\mathbb{C}_{a}$. These Grassmann numbers play the role of local coordinates of the super Poincare group $S\Pi$.      

Further, it can be shown that this superspace is an $S\Pi$-module, whose odd coordinates transform as Majorana spinors under the $SL(2,\mathbb{C}_{c})$ action. 

The transformations are given as follows

$x^{\prime a}=(\exp{K})^{a}_{b}x^{b}$,

$\theta^{\prime}_{\alpha}=\left(\exp(\frac{1}{2}K^{ab}\sigma_{ab})\right)^{\beta}_{\alpha}\theta_{\beta}$,

$\bar{\theta}^{\prime\dot{\alpha}}=\left(\exp(\frac{1}{2}K^{ab}\bar{\sigma}_{ab})\right)^{\dot{\alpha}}_{\dot{\beta}}\bar{\theta}^{\dot{\beta}}$,

$x^{\prime a}=x^{a}+b^{a}$,

$\theta^{\prime\alpha}=\theta^{\alpha}+\epsilon^{\alpha}$,

$\bar{\theta}^{\prime\dot{\alpha}}=\bar{\theta}^{\dot{\alpha}}+\bar{\epsilon}^{\dot{\alpha}}$,

for arbitrary $\left\{x^{0},x^{1},x^{2},x^{3}|\theta^{\alpha},\bar{\theta}^{\dot{\alpha}}\right\}$ and $\left\{x^{\prime 0},x^{\prime 1},x^{\prime 2},x^{\prime 3}|\theta^{\prime\alpha},\bar{\theta}^{\prime\dot{\alpha}}\right\}\in\mathbb{R}^{4|4}$.

From the above expression, we see that we can indeed view the odd part of the superspace as Majorana spinors.

The physical intuition behind these super-generalizations is as follows.

1. Usually, people would like to treat the classical bosonic fields as complex-valued function. We may also treat the classical fermionic fields as complex-valued spinors. In most textbooks, we consider the unitary irreducible representation of the Poincare group as the 1-particle state. Knowing the transformation rule of 1-P states under action of the Poincare group, it is possible to recover the field equation satisfied by the classical fields. This is shown in the paper "Unitary Representations of the inhomogeneous Lorentz Group and their Significance in Quantum Physics" by Norbert Straumann. 

http://arxiv.org/pdf/0809.4942v1.pdf

Following the procedure, we obtain complex-valued Dirac spinor fields. However, in quantum physics, taking the classical limit $\hbar\rightarrow 0$, the commutators (anticommutators)

$[\hat{\phi}^{I}(t,\vec{x}),\hat{\phi}^{J}(t,\vec{y})]_{\pm}=o(\hbar)$ leads to either Grassmann-even or Grassmann-odd classical fields defined as

$\phi^{I}(x)$: $\mathbb{R}^{4}\rightarrow\mathbb{C}^{p|q}$. 

In this classical picture, the canonical quantization is much clearer: we replace commutative Grassmann numbers by elements of the Weyl algebra, and replace anti-commutative Grassmann numbers by elements of the Clifford algebra.

2. The picture of complex-valued classical fields is correct as long as we consider only non-interacting free fields. Let us consider the classical electrodynamics, given by the following action 

$S=-\int d^{4}x\left\{F^{ab}F_{ab}+\bar{\Psi}{\not}D\Psi\right\}$,

the non-homogeneous equation of motion $\partial_{a}F^{ab}=\bar{\Psi}\gamma^{\mu}\Psi$ shows that the bosonic fields $A_{a}$ are propagated by the charges carried by fermionic 'bi-fields', while the homogeneous equation of motion $\partial_{[a}F_{bc]}=0$ tells us that the bosonic fields $A_{a}$ has free-dynamics. This inevitably forces us to re-consider about a better definition of classical fields. 

To show this, let us review the Grassmann algebra $\Lambda_{\infty}$ generated by infinitely many generators $\xi^{1}$,$\cdots$, satisfying $\xi^{i}\xi^{j}+\xi^{j}\xi^{i}=0$. Any element $z\in\Lambda_{\infty}$ can be represented as 

$z=z_{B}+z_{S}=z_{B}+\sum_{k=1}^{\infty}\sum_{i_{1}\cdots i_{k}}C_{i_{1}\cdots i_{k}}\xi^{i_{1}}\cdots\xi^{i_{k}}$,

where $z_{B}$, $C_{i_{1}\cdots i_{k}}\in\mathbb{C}$. We call $z_{B}$ the body of $z$ and $z_{S}$ the soul of $z$. 

Inspired by the above definitions, it is much clearer if we think of the classical bosonic field $A_{a}$ as Grassmann-even-valued fields, whose body as a $\mathbb{R}$-valued function propagates in spacetime freely and soul is propagated by the charge carried by the Grassmann even source.  

In the classical limit, one would also expect that all the spin degrees of freedom vanish as the planck constant $\hbar$ goes to $0$. The reason we treat classical fields as Grassmann-valued functions is to allow us to have a formal construction with a consistent classical action with interactions. To some extend, we should say that the electromagnetic radiation is purely a quantum effect, which is absent in the classical limit. 

This construction also explains why we use Grassmann-odd-valued spinors for the path-integral formalism of fermions. 

In this new picture of classical field theory, both classical bosonic fields and classical fermionic fields are unified as

$\phi^{I}:$ $\mathbb{R}_{c}^{4}\rightarrow\mathbb{C}^{p|q}$.  

In this new picture, the classical Dirac fields $\bar{\Psi}$ and $\Psi$ should be independent elements of the set of Grassmann-valued fields. As a result, it is inappropriate to define the relation $\bar{\Psi}=\Psi^{\dagger}\gamma^{0}$. But still, we can have the involution of the complex Grassmann algebra defined as 

$(zw)^{\ast}=(w)^{\ast}(z)^{\ast}$.

I am not sure how to find the rule of involution relating spinors and anti-spinors so that the classical Dirac Lagrangian is real. 

In addition, the bonus benefit of this picture is that it allows us to have the 'classical' mass term of Majorana fermions.

All the details can be found from the book  'Ideas and Methods of Supersymmetry and Supergravity or A Walk Through Superspace' by Ioseph L Buchbinder and Sergei M Kuzenko.

Please leave a comment if you have any idea how to show that the Dirac Lagrangian takes values in $\mathbb{R}_{c}$.    

answered Feb 15, 2016 by XIaoyiJing (50 points) [ revision history ]
edited Feb 22, 2016 by XIaoyiJing

Since this is supposed to be an answer, can you please give enough details so that it actually constitutes an answer your own question? Or was this supposed to be just a comment? 

@ArnoldNeumaier 
Actually, I am still working on this. I probably should have added this as a comment. I thought I could have worked it out very soon so that I will be able to finish my answer. This final answer is related with a theory called 'Grassmann-shell', which shows the isomorphism between a $\mathbb{Z}_{2}$-graded Lie algebra and a Berezin superalgebra. This theorem allows us to use the Grassmann-valued Majorana spinor as a $SL(2,\mathbb{C})$-module. This is exactly the construction of the $N=1$ Poincare superalgebra.

I found two references explaining this $Grassmann-shell$ theory. The first one is 'Foundations of Supermathematics with Applications to N=1 Supersymmetric Field Theory' by Cook, James Steven. The second one is 'Ideas and Methods of Supersymmetry and Supergravity or A Walk Through Superspace' by Ioseph L Buchbinder and Sergei M Kuzenko. 

Now, I still have to go through the whole chapters so that I will be able to totally understand the theory. After that, I will re-edit my previous answer so that it will be helpful for others who are interested in the question I posted. 

@ArnoldNeumaier 

I was confused by this for the following reasons. 

First, I realized that some people treat components of classical spinors as complex (or real) numbers, but some other people assumed that components of classical spinors anticommute. 

Then, after reading the papers by Berezin mentioned in my question, I learned that it is better to treat classical spinors as Grassmann-valued objects because their canonical quantization gives the fermions. In addition, if we treat components of classical spinors as Grassmann numbers, we can have a classical Lagrangian of massive Majorana fermions.

Now, let us follow the natural assumption that classical spinors should be Grassmann-valued. For the classical Weyl fields, we have spinors transforming as a $SU(2)$-module. This reminds me of the first paper by Berezin that I mentioned in my posted question. In that paper, Berezin showed that quantising the Grassmann algebra generated by three Grassmann numbers $\theta_{1}, \theta_{2}, \theta_{3}$ leads to the spin-1/2 Pauli matrices in quantum mechanics.

At this stage I became very confused. We can start from a three classical Grassmann numbers and quantize them to get $su(2)$ algebra in quantum mechanics. On the other hand, in the classical Dirac field theory, we want Grassmann-valued spinors transform under the $SU(2)$ group. 

Then, I was wondering if there exists any fundamental theory assuring that classical spinors should indeed be Grassmann-valued. However, when we are taking the classical limit $\hbar\rightarrow 0$, all the spin degrees of freedom vanish. My hope was that there might be a mathematical theory claiming that we can construct an $SL(2,\mathbb{C})$ module whose elements have Grassmann-valued components.

It turned out that there does exist such a constructive theory in mathematics. I realized that the super vector space has different meanings between some mathematicians and physicist. This difference may answer my question. 

For mathematicians, a super vector space is simply a $\mathbb{Z}_{2}$-graded vector space over $\mathbb{C}$ or $\mathbb{R}$. For physicists, the definition is a little bit different for they define it as a module over Grassmann algebra such that the odd part of any vector anti-commute with any a-number. 

Then, we can replace the above super vector space by a Lie superalgebra (in the sense of mathematician's definition), it has a corresponding Lie superalgebra over Grassmann numbers (which is called Berezin superalgebra). This Lie superalgebra over Grassmann numbers is the Grassmann-shell of the origal Lie superalgebra over $\mathbb{C}$.

I have to stop here today. Tomorrow, I will try to finish studying the theory of Grassmann-shell and confirm that it indeed can answer my question.       

If there is anyone who is still interested in this question, please be patient. 

After almost a week of study through the book 'A Walk Through Superspace', I finally got the answer. The book totally solved my confusions. The above comment I left about the 'Grassmann-shell' is closely related with the final answer to my question, but is not the final answer. My above comments contain incorrect information. 

Tomorrow, I will start posting the answer to my question. It will probably take me a long time to finish because the answer involves a lot of mathematical concepts about superspace and physical intuitions about classical field theory.

Previously, the mistake I made was that I didn't distinguish the differences between 'superalgebra', 'Berezin superalgebra (i.e. the Grassmann-shell of superalgebra)' and 'super Lie algebra (i.e. the even part of Berezin superalgebra)'. 

The final answer goes roughly as follows:

We can construct the Poincare superalgebra as a $\mathbb{Z}_{2}$-graded vector space over $\mathbb{C}$ with graded bracket. Then, we construct its Grassmann-shell, which is a $\mathbb{Z}_{2}$-graded vector space over the Grassmann algebra $\Lambda_{\infty}$. The even part of this Grassmann-shell possesses a normal Lie bracket (i.e. the un-graded Lie bracket). This even part of the Grassmann-shell of the Poincare superalgebra is called the super Poincare Lie algebra. 

Then, by exponentiation of the super Poincare Lie algebra, we obtain the super Poincare Lie group, which locally looks like the superspace $\mathbb{R}^{4|4}$. 

Finally, it can be shown that there is a natural left action of the Grassmann-shell of $SL(2,\mathbb{C})$ on the odd part of $\mathbb{R}^{4|4}$. In other words, we can lable the odd coordinate of $\mathbb{R}^{4|4}$ with spinor indices. The trick for showing this is similar as we show that the Minkowski spacetime can be viewed as a coset space of the Poincare group modulo the Lorentz group. 

From a physical point of view, this approach is more natural in classical field theory. For example, if we consider the Lagrangian of electromagnetic fields with interaction with electrons, then it is more natural to think of these fields as maps from superspace $\mathbb{R}^{4|4}$ to $\mathbb{R}^{4|4}$. 

I will try to explain these things clearly tomorrow or the day after tomorrow. 

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