Talk:Bispinor

Lorentz transformations

I'd like to see more information about how bispinors transform. Count Truthstein (talk) 16:52, 4 December 2012 (UTC)

The expressions for Lorentz transformations

I think that using $\chi$ for the boost parameter, and the $\phi ^{i}$ for rotation parameters is an extraordinarily bad choice. These two symbols are used for the lower and upper components of the bispinor $\psi$. If there is a hidden wonderful relation so that the formula (matrix multiplication $s[\Lambda ]\psi$) still makes sense, then this relationship surely deserves an explanation. YohanN7 (talk) 19:42, 7 January 2013 (UTC)

No, you're absolutely right. The source I got those expressions from didn't use those symbols for the upper and lower components. Count Truthstein (talk) 22:54, 7 January 2013 (UTC)

Cyclic definition

The first sentence of this article "Bispinor" reads: "In physics, bispinor is a four-component object which transforms under the (½,0)⊕(0,½) representation of the covariance group of special relativity". In that sentence, the expression "(½,0)⊕(0,½) representation" links to Representations of the Lorentz group#Common representations which states: "(1/2,0) ⊕ (0,1/2) is the bispinor representation (see also Dirac spinor)", with "bispinor" linking back to this article.

Quite unsatisfactory. (And Dirac spinor too is of no help as it links back to "bispinor" in its definition...) --Chris Howard (talk) 21:17, 8 January 2013 (UTC)

I don't think there is a big problem here. There would be a problem if you didn't understand what a representation was but I don't know if that's the problem. Simply, in order to say what a bispinor is, we need to say (a) what kind of mathematical object it is (in this case a four-component object) and (b) how it transforms with changes of the frame of reference. (Compare the definition of tensor which also includes transformation rules. ) Maybe we can also include examples, such as the solutions to the Dirac equation.

In more detail, a change in the frame of reference is represented by a Lorentz transformation, which is an element of the "covariance group of special relativity", or the Lorentz group as it's more commonly called. All it means for the object to belong to a representation of this group means that we have a rule for combining the transformation with the object to come up with a new object (plus extra obvious and useful criteria such as the ability to add these objects together (a vector space axiom) and that applying the transformation rules corresponding to two Lorentz transformations one after another has the same effect as composing the two transformations and applying the rule corresponding to the single resulting transformation (that's the homomorphism property which occurs in the definition of a representation)). This article for the most part tells you what these rules are (any Lorentz transformations can be obtained as a composition of translations (leaving the objects unchanged), boosts and rotations). It's not necessary to understand the classification of representations of the Lorentz group to understand this (e.g. I don't understand what "(½,0)⊕(0,½)" means yet). Count Truthstein (talk) 00:30, 9 January 2013 (UTC)

A self-referential definition of this kind simply makes no sense. Sure, the first sentence could probably better refer explicitly to the Lorentz transformation, but that's not the point. The problem lies in the fact that a definition of the type "An XX is an n-component object which transforms under the XX representation of the Lorentz group" does not define what XX is. No more and no less. --Chris Howard (talk) 19:51, 9 January 2013 (UTC)

Then it's a description, not a definition. I agree it's a problem that it looks like a definition. I'm not sure what should be done about this. Count Truthstein (talk) 23:12, 9 January 2013 (UTC)

I've tagged it for cleanup, having self-referential or lacking definition. --Chris Howard (talk) 07:19, 11 January 2013 (UTC)

Thanks for pointing to a minor problem with Representation theory of the Lorentz group, but this is not a cyclic definition. Try to understand what “(1/2,0) ⊕ (0,1/2) is the bispinor representation” means – the “bispinor representation” is the representation which is classified as (1/2,0) ⊕ (0,1/2), and a "bispinor" refers to an element of space where it acts. It is not a definition of (1/2,0) ⊕ (0,1/2) classifier from "bispinors", obviously. Incnis Mrsi (talk) 08:49, 11 January 2013 (UTC)

Just on the form, it is surprising that you don't see any issue with looping (linking) back onto the same term from within a definition. The loop is still in place, even now that you have improved the Representation theory of the Lorentz group article. It is possible to write a bispinor definition without the use of the term bispinor in its defining portion: an example is on page 135 of 978-3764362027. --Chris Howard (talk) 18:08, 11 January 2013 (UTC)

What looping are you speaking about? "Representation theory…" does not rely (and did not rely) on the "bispinor" article. If you are not happy with reduction of "bispinor" to "Representation theory…", still poorly written, then add your definition, at last. You claim you have an English book with it, but I have not. Incnis Mrsi (talk) 22:41, 11 January 2013 (UTC)

The "loop" is what I described what the reader effectively does when he tries to get a definition of a bispinor from Wikipedia; I wrote (at the top of this section):

The first sentence of this article "Bispinor" reads: "In physics, bispinor is a four-component object which transforms under the (½,0)⊕(0,½) representation of the covariance group of special relativity". In that sentence, the expression "(½,0)⊕(0,½) representation" links to Representations of the Lorentz group#Common representations which states: "(1/2,0) ⊕ (0,1/2) is the bispinor representation (see also Dirac spinor)", with "bispinor" linking back to this article.

That is, in practical terms the reader of Wikipedia, searching for a definition of bispinor (assuming that the reader knows mathematics, linear algebra and all that and knows about Lorentz transformations but has never heard of the word "bispinor") will see, from the first sentence of the "bispinor" article, that he has to understand the definition of "(½,0)⊕(0,½) representation" before he can understand the definition of bispinor. So the patient reader will follow that link and will carefully read what is written there. The link leads right into the middle of the "Representation theory of..." article, to the sentence "(1/2,0) ⊕ (0,1/2) is the bispinor representation (see also Dirac spinor)". The poor reader will be aghast to see that an understanding of the word "bispinor" which he has not understood appears to be a prerequisite to the understanding of "(½,0)⊕(0,½) representation", because in "bispinor representation" that term "bispinor" just links back to the same article. That is, he makes no progress at all if he has no prior knowledge of what a bispinor or a bispinor representation is.

I am sure that the lead of this article (or a separate "Definition" section for it) can be re-worded in a manner that avoids reader frustration in this sense. Had I a perfect definition at hand, I would have added that definition myself to the article - but as I do not, I raised the issue on the talk page, trusting that one of the authors of this article would rather easily re-word the definition. Regarding the book I gave as example, it is: Daniel Elton, Dmitri Vassiliev: The Dirac equation without spinors. In: Jürgen Rossmann (ed.); Peter Takac (ed.); Günther Wildenhain (ed.) (18 October 1999). The Maz'ya Anniversary Collection: Volume 2: Rostock Conference on Functional Analysis, Partial Differential Equations and Applications. Springer. pp. 133–152. ISBN 978-3-7643-6202-7. Retrieved 12 January 2013. {{cite book}}: |author1= has generic name (help), see page 135. The definition is: "a bispinor in [... four-dimensional Minkowski space] is a set of four complex numbers [...] which change under Lorentz transformations in the following way [... ...]". Looks like a better start to me. But I would really rather leave it up to the authors of the bispinor article themselves to decide on the exact re-wording. Hope this helps, --Chris Howard (talk) 08:40, 12 January 2013 (UTC)

Some small issues

Hi!

I think that the cleanup tag well could be removed. But sure, there are things to do.

• The property of being a bispinor does not, and should not, fix precisely what the object having the property is. Just as the definitions of "group" or "vector space" does not fix what group elements or vectors are. But I think user Chris Howard still has a bit of a point here. This could easily be resolved, since, for instance, a Dirac wave function is a bispinor at every point in spacetime. We could also say that every 4-dimensional complex vector space endowed with the (½,0)⊕(0,½) representation of the Lorentz group is a bispinor.
• The inline LaTeX should be replaced with html according to the standard guidelines.
• "A bilinear form of bispinors $\psi ^{\dagger }\otimes \psi$ can be reduced to five irreducible (under the Lorentz group) objects:" This is problematic. What is $\psi ^{\dagger }\otimes \psi$ doing here? This is notation I haven't seen. Besides, the dimension of the bilinear covariants (another term for the thingies) could be listed. Clifford algebra should be linked.
• I have plans for including an outline of a derivation of the (½,0)⊕(0,½) representation (the matrix S in this article) somewhere. Should it go here or should it go to Representation theory of the Lorentz group? Should it, for that matter, go anywhere? Perhaps it would be overkill, but some readers probably want to know how it comes about.
• As above, but for object transforming the antisymmetric tensors.
• "Here the physical point with coordinate $x$ is moved to have a coordinate of $\Lambda x$, and so the old value at $\Lambda ^{-1}x$ tells us what the new value at $x$ is." This is really problematic. I'm sure that the intention when writing it was correct, but it doesn't read right, at least not for me. There is one and only one point in physical spacetime involved. This one physical point p is called x before we make the transformation. After the transformation the same physical point p is called Λx. In other words, in the two sentences plus the equation describing the transformation, x, Λx and Λ^-1x all refer to the same point p in spacetime. We need a reformulation here, or we can keep the equation only and leave the interpretation to the reader. YohanN7 (talk) 23:00, 17 February 2013 (UTC)

— Preceding undated comment added 00:17–07:06, 18 February 2013 (UTC)

If “bispinor” is an article, not a {{mathdab}}, then it should specify which objects are covered. The analogy with groups is inappropriate: there is an immense amount of groups with a plenty of additional structures; the same about vector spaces. About bispinors, we have either complex or real 4-dimensional vector space… did I miss something? I do not see a major problem here. IMHO the main problem with explaining concepts originated from physics in mathematical language is the ubiquitous confusion between algebraic spaces and wave functions or fields with corresponding values (the latter being sections of spacetime-based bundles). Also, it is ironical that you speak about de<math>ification of body text (BTW not referring to any relevant guideline explicitly), but use "$x$" in your post of 00:17 (UTC). Changes from/to <math> are the least priority, if any. If you want to make useful tweaks with formulae, look at WP:MOSPHYS #Super-/sub- scripts in nomenclature (unofficial) instead. Incnis Mrsi (talk) 17:46, 18 February 2013 (UTC)

• Vector spaces are more than analogy. You point out yourself above that a bispinor is a 4-component vector, real or complex, while the article talks about a 4-component object.
• The article talks about the space of bispinors, as if there was only one. I want to define bispinor somewhat precisely algebraically, and then later exemplify (for example, the XXX is a bispinor field...). This procedure also takes care of the problem you mention about the confusion involving sections of bundles. In other words, bispinors often do live in spaces with plenty of extra structures. [This is your point (and mine), which is clashing with a previous point of yours about extra structure in the same post above.]
• When it comes to deriving the matrix S, then I can only imagine that you are jumping up and down with joy since you left that without comment/and or pun. Correct? YohanN7 (talk) 12:27, 19 February 2013 (UTC)

I may be wrong about the LT description. If the interpretation is that of an active transformation, then there really are two distinct points in spacetime involved. In this case one and only one coordinate system is involved. My bad. YohanN7 (talk) 16:45, 19 February 2013 (UTC)

Re definition of bispinor, there aren't many 4-dimensional complex vector spaces endowed with the (½,0)⊕(0,½) representation - that's close to a unique definition apart from various changes of basis. (I think I agree with what Incnis Mrsi said about this above.) I don't know what the significance of the bilinear form is either. I've replaced the explanation of the transformation rule, although it is confusing in a subtle way - it writes \phi(x) to mean the function \phi. We are not taking an individual value \phi(x) and transforming it under some rule, but replacing the value at x with a value based on a completely different point. Count Truthstein (talk) 19:07, 21 February 2013 (UTC)

The significance of the listed bilinear forms is that they are the only bilinear forms on V×V for V a space of bispinors that have well behaved Lorentz transformation properties. By extension, this holds too for bispinor fields. They are thus suitable ingredients in a Lagrangian, perhaps contracted with ingredients coming from other fields, that also have well behaved LT properties. The Lagrangian is required to be Lorentz scalar. The way to show this is to use the displayed transformation rule Ψ→SΨ together with an expression for S (and its properties) and properties of the Dirac algebra. (That article is in serious need of beefing up. Perhaps this should go there.)

I suppose the LT description is fine.

Now for the definition. We don't disagree substantially, even if Incnis manages to put it that way. The property of being a space of bispinors is an algebraic one, which is exactly, as you say, pinned down by being a complex vector (a real one will not do) space transforming under the (½,0)⊕(0,½) representation. Where we potentially disagree is only in the usage of the space of bispinors, as opposed to a space of bispinors. For one thing, (½,0)⊕(0,½) representations on a space V of the Lorentz group become matrices only after a choice of basis for V. For example, in the smooth manifold setting, the Lie algebra of the Lorentz group is a set of left invariant vector fields on SO(3,1). The elements are differential operators on the set of smooth functions on SO(3;1), not matrices. The uniqueness of (½,0)⊕(0,½) representations applies only up to equivalence of representations, which may amount to more than a change of basis. For another thing, spaces of bispinors are used as ingredients in the construction of more complex objects, see e.g. Bargmann-Wigner equations. Suggestion:

(Present text):In physics, a bispinor is a four-component object which transforms in a specific way under Lorentz transformations: specifically, the space of bispinors is the (½,0)⊕(0,½) representation of the Lorentz group[1] (see, e.g., [2]). Bispinors are used to describe relativistic spin-½ quantum fields.

(Suggested text):In physics, a bispinor is an element of a 4-dimensional complex vector space which transforms in a specific way under Lorentz transformations: specifically, a bispinor is an element of a (½,0)⊕(0,½) representation of the Lorentz group[ref 1] (see, e.g., [2]). Bispinors are, for example, used to describe relativistic spin-½ wave functions.

The suggested change of the example (quantum field->wave function) is motivated by the fact that in quantum field theory, the fields refer to field operators (operators on Hilbert space) rather than elements of the Hilbert space (state vectors represented by wave functions). The quantum field transform according to different rules. YohanN7 (talk) 23:44, 21 February 2013 (UTC)

This suggested change has the added bonus that all circularity (real or imagined, see earlier posts) is removed. YohanN7 (talk) 23:53, 21 February 2013 (UTC)

One final point: The suggested change would make the definition exactly parallel the conventional physics definitions of scalars, vectors pseudo-vectors and tensors. Those definitions do not speak of the tensor field or the like. YohanN7 (talk) 00:14, 22 February 2013 (UTC)

The suggestion looks good to me. It is a very minor point but it would be possible to remove the definition of vector space as a prerequisite for getting benefit from it. How about

(Suggested text):In physics, a bispinor is an object with four complex components which transforms in a specific way under Lorentz transformations: specifically, a bispinor is a member of the 4-dimensional complex vector space considered as a (½,0)⊕(0,½) representation of the Lorentz group[ref 1] (see, e.g., [2]). Bispinors are, for example, used to describe relativistic spin-½ wave functions. (Count Truthstein)

I am happy with that if we can change one little word:

(Suggested text):In physics, a bispinor is an object with four complex components which transforms in a specific way under Lorentz transformations: specifically, a bispinor is a member of the a 4-dimensional complex vector space considered as a (½,0)⊕(0,½) representation of the Lorentz group[ref 1] (see, e.g., [2]). Bispinors are, for example, used to describe relativistic spin-½ wave functions.

This is extremely minor, but it is this that enables us to remove that annoying cleanup-rewrite request. Moreover, I'll add a very explicit definition (transformation rule) of a (½,0)⊕(0,½) representation to either this article or to Representation theory of the Lorentz group. Cheers! YohanN7 (talk) 04:04, 23 February 2013 (UTC)

With that change (good job!) indeed the cleanup tag can and could go. --Chris Howard (talk) 21:56, 23 February 2013 (UTC)

Proposal

Hi!

I have written a very explicit description of (one type of) bispinors. It is the first four sections (including "spinors introduced") of this: User:YohanN7/Bispinor. There is more in the proposal, but I am not sure whether this should go here or in the Dirac algebra article. I think maybe the latter is best.

I am aware of some errors, bugs, and lack of references, and one total omission. The omission is a lack of explicit choice of gamma matrices, and therefore the lack of an explicit expression for S(Λ). It is a piece of cake to fix those problems, but it is quite time consuming. I need some input before I proceed, so that I don't work in a vacuum.

The bispinors described are quite different from the spinors described in the spinor article. The latter sit in the Clifford algebra, and are concretely matrices, while the ones I describe are complex column vectors, but both variants are bispinors in the abstract sense.

There is one headline missing in the present article, and that is "Majorana spinors". From what I understand, these are ordinary bispinors restricted by the further condition that the representation of the Lorentz group can be chosen real in a certain Lorentz invariant sense. This means that the gammas are chosen purely real (or purely imaginary if the metric is (-+++)), and the vectors on which they act are constrained by an equation (Majorana equation). Majorana spinors are still bispinors, because they transform under the (1/2,0)⊕(0,1/2) representation and nothing else. But this means, of course, that the definition of bispinors must allow for real vector spaces as well as complex, so I was wrong in a post above. (In practice, in QM, the Hilbert space is still by definition complex. The Majorana representation means that it is possible to chose a purely real Ψ to represent a state that stays real under LT) YohanN7 (talk) 21:44, 2 March 2013 (UTC)

After a super-rapid skim of User:YohanN7/Bispinor it looks reasonable. Much or most of it could be merged into here. It would be a bit of a chore as this article is not terribly well organized. As to Majorana, I'm hacking on Majorana equation right now. It's a chore. However, you ain't seen nothin' til you've seen Elko. I'm trying to write Draft:ELKO Theory ... but whuff. Pretty much all Elko references are horrible. It seems to be Majorana with a twist, and I'm starting to see what the twist is, but the primary sources are annoyingly naive and filled with algebraic errors. Anyway, I think Dirac spinor is in pretty good shape and I want to get Majorana spinor up to a comparable condition. 67.198.37.16 (talk) 04:28, 30 November 2020 (UTC)

Edit 2013-03-09

Done:

• New section: Derivation of a bispinor representation
• Removed $\psi ^{\dagger }\otimes \psi$, which was probably originally a typo-like mistake.
• New reference: Weinberg

To do:

• Small edits needed so that old and new material tell the same story. (E.g. specify metric and other choices everywhere such a choice is made.)
• The Majorana representation needs a section of its own.
• References are made to the Dirac algebra. This article is not yet beefed up with the right stuff. I have some in preparation.
• "The Dirac basis is the one most widely used in the literature." Appropriate for low energy phenomena. Needs ref.
• Expressions for Lorentz transformations of bispinors|Expressions for Lorentz transformations of bispinors should contain both active and passive transformations.
• "...where Λ is a Lorentz transformation" Add "and S its bispinor representation."
• " commutation relations in (C4) are exactly those of so(3;1" Add "given by M1"
• Unlink second mention of boost parameter and link first.
• {{math|U×U} -> U×U} in Dirac algebra
• Need to allow for real vector spaces in def due to Majorana representations (as opposed to Majorana spinors which like all QM fields by def is complex.).
• See also need several new entries.
• My use of inline citations is apparently deprecated. YohanN7 (talk) 17:31, 9 March 2013 (UTC)

I'll wait a day or two with any finishing touches. YohanN7 (talk) 16:58, 9 March 2013 (UTC)