Talk:S-matrix

It would be very helpful if there was a discussion of scattering parameters from the point of view of a microwave or RF engineer. I would be willing to provide such an explanation if others think it would be useful. Mradway 21:17, 25 Mar 2005 (UTC)

I would be interested in hearing what you have to say about the relation between classical field theory scattering operators and the quantum field theory version. Bas Michielsen

I think the point of view of a microwave/RF engineer would be very useful, and I encourage Mradway to provide this explanation. 05:48, 16 June 2006 130.188.8.11

Wow, my comment is essentially already here: We need a discussion of the scattering matrix formalism; there's really no particular need to have this topic described in a manner narrowly centered on quantum mechanics (and even more narrowly, particle physics). I understand it's even used in quite "classical" electrical transmission lines, and reading Siegman's book "Lasers", I realize that this earlier usage must be resp. have been widespread enough for him to omit the discussion e.g. of concatenation or free space evolution of radiation. So please go ahead, Mradway, your contribution is exactly what I came looking for! -- anonymous, July 10th 2007

Hey you guys should check out s-parameters where this topic is described in great length. 128.138.200.94 (talk) 21:15, 15 January 2010 (UTC)

One sees complex conjugation is involved. A generalization to 'line-based exponentials' (that is e^(M.x) where M are some matrices). The rule for that is GAG^(-1)=W*A, where W is the symmetry matrix (basically the linear map describing how a generalized conjugate works on the lie group parameters, and G is some kind of distance). Now if one wants this to work, one decomposes the wavefunctions into lie-group plane waves (or pseudo-lie group plane waves using a squaring law, rather than a product of two numbers, and using linearity.)(alien-like signals, perhaps (with multiple pseudo-time dimensions labeled by x)), and one must detect the W matrix being applied. The best idea to try is SMAT(wf,e^(M.(nq))), for arbitrary matrices N. Now one calculates the possible M's and solves the equation GMG^(-1)=W*M, and calculate W. W is then the group operations describing what conjugation really means. The interesting thing about this kind of conjugation, is the domain of G can be tuned to inner automorphisms when it is the same domain as the lie group from the lie algebra M. Changing the domain of G, gives some nice control features. The inner automorphisms are like 'super-conjugates' as a easier name, it is like having the complex conjugation operator as a 'number' along with sqrt(-1). Those conjugates are very important for studying the symmetries regarding a squaring law using carefully choosen SU(n) matrices. Depending on the choice, one can likely realize squaregraph symmetries (for my post-quantum theories) OR one could be humble and use it for quantum scattering with square-law lie-group based plane waves. One might even vary the plane waves over space and time, causing complete chaos in this approach, which shouldn't be feared. Most Mathematicians fear chaos, but that is the point. Lots of choices mean lots of applications and most of all more learning. The example I saw used conjugation, so I'd better tell people what conjugation really is, rather than being restricted to complex numbers. Complex theory is excellent, but those days are semi-over, and results regarding conjugation should be examined like the world is about to end, because there is a lot of gold in the notion of conjugation. inner automorphisms are already discovered for lie group, but one is unable to control the tangent space in that theory. As if lie groups are places, one has to eventually decide a frame of reference for the tangent space, which observations might eventually differ. For current theories everything works, but SU(n) with a squaring law associated to it behaves 'differently' depending on the choice of generators. That applies when one considers a time-integral as equivalent to a multi-displacement-integral with a rank 3 matrix contracting the object being integrated so where G(dx/dt,d/dt)=dx/dt*d/dt is the alternative description, and G need not be a closed group, but weirdly it is like a group but not closed. Numbers begin to behave differently, and that includes all the lie groups. The concept of free will (true or not, likely both true and not true depending on who tests it, y=y is different than y=x, thus observers feel something similar to free will due to reflexivity of their self-compared outcomes. ) makes theories richer when one considers all possible actions a person or system could take and one learns about the universe far more quickly. Free will is an indepensible mathematical toolbox, and sensible ambiguity saves time (and possibly everybody's future money). ~2026-29937-4 (talk) 20:43, 14 January 2026 (UTC)

Also it'd be nice to consider non-linear operators as being similar to lie group matrices. One can redefine the integral. One uses int_t:=(bounds(0,t)(int_x*G)), where G is similar to rank 3 (a 3D matrix if needed), but can also include non-linear operators. One then gets crazy and can redefine the exponential map. Now one changed the definition of a circle (but unfortunately in a coordinate dependent way). A coordinate independent generalization would be nice. Gauge fixing? I tried gauge fixing a non-linear operator, by I:='inverting it',A:='abelianizing it' (via ordering the path motion, assuming L-shape) and the formula is AIAIA, which failed on x^(1/4) because the ordering of the function restricted the domain so inversion becomes undefined in finite time, thus it fails to work properly. I tried locally applying it (via small rectangles, brick by brick for all paths), but it is hard to find meaning in that, AND near a x^(1/4) defect it has no meaning no matter how small the bricks are. Another avenue is a pseudo-tensor (perhaps the Landu-Lipschitz), but it doesn't seem likely that it is a single frame, and more proof is needed, and that article is unclear about that. More problems arise requiring that the space that is smooth remains smooth, which would allow all differential equations to be invariant in the gauge fixed regime. Also I wonder if the pseudo-tensor relies too heavily on a global metric and the einstein field equations, which I want gauge fixing for all time-parameterization-independent path map operators (possibly with a start time) (that is (s(t)->x)->(s(t)->y)) and one wants a new map that gauge-fixes deformation, which I would define as a function that is an inverse function. Applying the gauge fix should 'fix' the deformation-function applied to y, and yield a unique result. Then general relativity and curvature tensors wouldn't be required for the principle of covariance and newer theories of gravity can be created quickly. ~2026-29937-4 (talk) 20:58, 14 January 2026 (UTC)

Would it be useful to move the stuff about Wick's theorem to another page? --HappyCamper 16:05, 4 September 2006 (UTC)

I agree with this. Wick's theorem is far more general than something that is used to evaluate S-matrix elements. PhysicsBob 09:19, 26 July 2007 (UTC)

I copied the section there (it was a redirect here) as a start on that article. Now it alos needs some work and this section can be trimmed. RJFJR 13:48, 2 November 2007 (UTC)

The article at present seems to read like a page of mathematics rather than an encyclopedia entry. The LSZ reduction section especially reads like a mathematical proof. Perhaps someone who is competent in these matters could insert more explanation about the underlying ideas and applications of the S-matrix. Unfortunately, I don't know much about this kind of thing. PhysicsBob 16:19, 26 July 2007 (UTC)

The LSZ reduction formula has its own article; is there any reason why it needs a separate section here? --Starwed 19:25, 29 October 2007 (UTC)

I feel there is no reason why the section on the LSZ reduction and Wick's theorem should remain in the main article, since they are techniques used to evaluate S-matrix elements, rather than being part of the defintion of the S-matrix. Perhaps a 'see also' section would be more appropriate? PhysicsBob (talk) 06:45, 11 June 2008 (UTC)

Towards the end of "mathematican definition" there is a list of properties that the S matrix "must" have. There is no mathematical need for these, as I understand it, but they are rather connected to (admittedly very reasonable) physical properties. like: unitarity <--> conservation of probability does not change vacuum <-- energy conservation <--> time independence of lagrangian S|k> = |k> <--> stable (non-decaying) particle momentum conservation <--> space independence of lagrangian most of these are related to Noethers theorem, which could be linked to. In fact, most of these are properties of the lagrangian rather than properties of the S-matrix imo, although they are tightly connected. Not really sure how to organise this article... Art.cascade (talk) 16:50, 25 November 2011 (UTC)

In the paragraph "Use of S-matrices", it seems to me that is is not the "Heisenberg" picture, but the "interaction" picture, isn't it ? — Preceding unsigned comment added by 31.33.159.210 (talk) 13:03, 26 July 2012 (UTC)

Weinberg describes things seemingly very differently. The in-states and the out-states inhabit the same Hilbert space. The scattering matrix is then the set of inner products of suitable basis vectors (well defined free multi-particle like Lorentz transformation properties) between them. These states are eigenstates of the full Hamiltonian. Then there is the S operator. This is defined such that it's matrix elements between free particle states equal the S-matrix elements. I.e. <b_free|S|a_free> = S_ba = <b_out|a_in>. [This presupposes a suitable subdivision of the Hamiltonian in free and interacting parts.] Is this the same thing as described in this article? YohanN7 (talk) 14:07, 28 January 2013 (UTC)

At any rate,

${\displaystyle \left\langle F,\beta |I,\alpha \right\rangle ,}$

(from article) doesn't make sense unless the states inhabit the same Hilbert space. YohanN7 (talk) 14:19, 28 January 2013 (UTC)

There is indeed an incoherence: if we say that S takes us from ${\displaystyle {\mathcal {H}}_{IN}}$ to ${\displaystyle {\mathcal {H}}_{OUT}}$ then ${\displaystyle \left\langle I,\beta |S|I,\alpha \right\rangle }$ makes no sense.

I am Dibyendu Bala,student of Tata Institute of Fundamental Research,want to extend this page in the following way.I want to add S-matrix in one dimensional quantum mechanics for a plane wave function(definite momentum state).Initially I will prefer to give the definition of S-matrix in case of short range potential barrier.Then relate transmission coefficient and reflection coefficient in terms of matrix element of S-matrix.I am also interested to show that the unitary matrix property is related to the conservation of probability current.Finally I will also include the effect of symmetry of the Hamiltonian on the S-matrix and the one dimensional version of optical theorem.

I like the new section. The problem is that there are errors. The k to the left and right of the origin aren't identical. The potential V(x) should be given explicitly, and inline LaTeX shouldn't be used for inline equations. Please fix them. Then, also, expressions for the S_ij wouldn't be out of place. Moreover, make sure to include inline citations. (I suspect the added reference is there for this purpose. Also, use the citation template for references (as for the other references.) YohanN7 (talk) 01:50, 26 October 2014 (UTC)

You need to note that this is (I guess) not the S-matrix in the context of quantum field theory, but rather from ordinary quantum mechanics. Citations are still lacking. YohanN7 (talk) 13:03, 27 October 2014 (UTC)

Yes,in this new section S-matrix is in ordinary quantum mechanics.Dibyendum (talk) 16:40, 27 October 2014 (UTC)

Thanku YohanN7 for correcting this new section.Dibyendum (talk) 16:40, 27 October 2014 (UTC)

You are welcome. (Please sign your posts and indent them properly. Type four tildes →~ to sign.)

You need to make it clear that this is ordinary QM in the article. Also, since this is an "easy" example, it should probably be one of the first sections. (Don't forget inline citations.) YohanN7 (talk) 14:09, 27 October 2014 (UTC)

The k to the left and right of the origin aren't identical. When I wrote that I was under the impression that a Step potential was to be treated. My bad, so forget that. YohanN7 (talk) 07:55, 2 December 2014 (UTC)

It looks like the odd conventions of this section, namely S→symplectic metric, instead of the identity, for the free theory (T=0) may well confuse the reader who has just read that S→I in the preceding sections.

Specifically, assuming a universal time-dependence by a phase exp(−iωt) in ψ, A describes an in right-mover and C an out-right mover; while D an in left-mover and B an out left-mover. So, interchanging B with C in ψ_out while leaving ψ_in alone, has the effect of interchanging the two rows of the S-matrix written at present, so that the free case is, indeed, the identity, as advertised above this section.

In this way, the optical theorem could present less deformed, indeed, bizarre, and the missing further relation for the imaginary part of r would be apparent. Cuzkatzimhut (talk) 02:03, 2 December 2014 (UTC)

But this convention is used in Eugen Merzbacher quantum mechanics book.Dibyendum (talk) 09:13, 2 December 2014 (UTC)

(It's common practice to indent replies.) This is the perfect example of where you need to put in a literature reference as a footnote. I'll do this one. YohanN7 (talk) 09:34, 2 December 2014 (UTC)

Done. Feel free to fill out with page number or anything. The ref-tag creates a popup, and the template harvnb connects to the reference list and places the actual popup text under "Notes". It takes a while to get used to the system, but it works. YohanN7 (talk) 09:45, 2 December 2014 (UTC)

OK, for future reference by the inquiring reader, but not for the article, I record that S₂₂ = 2ir* (1+2it)/(1−2it*), as things stand defined at present. Cuzkatzimhut (talk) 16:24, 2 December 2014 (UTC)

I think the new section should be moved closer to the top since it is relatively accessible. Some reorganization is required for this. But I don't know exactly what that reorganization would be. Just an idea. YohanN7 (talk) 08:06, 2 December 2014 (UTC)

Done. Looks ok? YohanN7 (talk) 12:24, 2 December 2014 (UTC) Yes,its looking ok.Dibyendum (talk) 12:27, 2 December 2014 (UTC)

Sanity-check for the intro to the new section appreciated. I think the "continuous matrix" can be formalized rigorously, but that would be gross overkill to do here. YohanN7 (talk) 15:26, 2 December 2014 (UTC)

The comment(s) below were originally left at Talk:S-matrix/CommentsTalk:S-matrix/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Last edited at 01:50, 1 January 2012 (UTC). Substituted at 05:09, 30 April 2016 (UTC)

Should the "S" in "S-matrix" be italicized. Usually mathematical variables (that do not spell a word) are written in italics. Should this apply to all "S-matrix" in this article?--ReyHahn (talk) 16:28, 28 June 2021 (UTC)

Looking at some of the sources the last two, Mussardo (1992) and Zamolodchikov (1979) both have the S in italics. The others are books I don't have access to. --Salix alba (talk): 18:40, 28 June 2021 (UTC)

The Peskin and Griffiths too, I will proceed to make the changes soon.--ReyHahn (talk) 19:25, 28 June 2021 (UTC)

Expressions like these (in the subject) make articles not very trustworthy. Is someone trying to sell a product here???

I removed the cleanup tag regarding the proof for unitarity for two main reasons. 1) It is not true that proofs require prose. See for example advanced graduate courses in pure mathematical logic. 2) I provided sufficient context preceding the proof so that the symbols can be read unambiguously, allowing the reader to make the correct conclusion. MMmpds (talk) 20:37, 25 November 2025 (UTC)