It finally has occurred to me that there is a similarity between i = √-1 and √2. They are each linearized representations of essentially quadratic values. We use the former in complex numbers and include the latter in the real number system as an irrational number. Each has proved valuable and is part of accepted mathematics. However, an irrational number does not exist as a linear value because it is indeterminate – that is what non-ending, non-repeating decimal number means: it never can exist. Perhaps we need an irrational number system as well as a complex number system to be rigorous.
The sense of this observation is that some values are essentially quadratic. An example is the Schrödinger Equation which enables use of a linearized version of a particle wave function to calculate the probability of some future particle position, but only after multiplying the result by its complex conjugate to produce a real value. Complex number space is used to derive the result which must be made real to be real, i.e., a fundamentally quadratic value has been calculated using a linearized representation of it in complex number space.
Were we to consider √-1 and √2 as similarly non-rational we may find a companion space with √2 scaling to join the complex number space with √-1 scaling along a normal axis. For example, Development of the algebraic numbers a + b√2 could include coordinate points with a stretched normal axis (Harris Hancock, Foundations of the Theory of Algebraic Numbers).
A three-space with Rational – Irrational – Imaginary axes would clarify that linearization requires a closing operation to restore the result to the Rational number axis, where reality resides.
[Note: most people do not think like I do, and almost everyone is happy about that: please read openly, exploringly, as if there might be something here. (Yes, my request is based on experience!) Tens of thousands of pages in physics and mathematics literature from popular exposition to journal article lie behind this inquiry, should you wish to consider that.]
Too much noise, too little signal. The set of numbers, a+bi, with a,b real, form a field, a set that's closed under addition and multiplication. Similarly the set of numbers a+bsqrt(2), with a and b real, form a field.
(In fact they define so-called extensions over the real or integer numbers, if and b are real or integer numbers.)
The fact that i and sqrt(2) are solutions of quadratic equations enters the picture when multiplying two elements, since the squares of i and sqrt(2) can then be seen to be elements of the field.
It is, however, possible to define other field extensions over the integers and the reals, using elements that aren't solutions of quadratic equations. This is, now, standard material on a course in algebra.
Cf. for instance https://faculty.etsu.edu/gardnerr/4127/notes/vi-31.pdf
Too much confidence, too little comprehension. It is comforting to know that mathematicians are alive and well, and with two echos already, good show!
Of course there exists jargon to set my inquiry aside. However my question addresses the condition that while the Schrodinger equation is useful to calculate probabilities, the result is not real until its true quadratic character is restored via multiplication by its complex conjugate. That the square root of two has similar character occurred to me along with possibilities of clarifying mathematics in a new light. I think that matters.
I am pleased for you that you have not been disturbed by curiosity.
I dislike the tone or our exchange, but I can do it too.
Happy Trails, Len
What is linear abouut Schrodinger is the operator (Hamiltonian) H , not wave function, which permits the superposition of solutions.
ie. in
H psi = E Psi
ie, if psi 1 and psi 2 are solutions , so is
a psi1 +b psi2
psi1 and psi 2 are complex functions , a and b complex numbers.
Any or all these could be accidentally real.
Stam is just giving you little exaples of abstract algebra, it sometimes comes in handy. The abstract point of view helps with trying to visualize irrational numbers, and completly abstract entities as i
You can use C without i as just (x,y) entities and give the rules. i becomes (0,1)
Howdy Juan Weisz,
Thank you for the pleasant reply and the information therein. I also appreciated Stan's information and only ruffled over the tone. Having read through several feet of my math shelf a few decades ago, and especially having gotten into Harris Hancock's Foundations of the Theory of Algebraic Numbers to his treatment of a + b(sqrt 2) I found Stan's contribution, and yours, quite clear. That development of algebraic numbers triggered what I consider a deep insight into my own efforts to deal with turbulent fluid dynamics so I didn't complete the book. Your comments on the Hamiltonian, superposition, complex numbers and abstract algebra also all ring true.
However, my question addressed a deeper curiosity which I tried to present in my comments there. Even using the Hamiltonian one is operating with a linear representation that provides an answer only after it is made real, and then it is only a probability! Just like back in '66 when engaged in such a discussion, I do not fault the rigor of what is accepted "to be mathematics" with other math-like issues undefined. That rigor is imperative for normal uses, including pure research into unreal abstractions that may be there. I am a natural philosopher, however, and what is happening in many mysterious methods that enable calculation interests me - thus my question. Do the methods tell me anything about the processes of nature that they enable us to calculate. (By the way, back in '66 I was told I had weak mathematical intuition despite agreeing as just stated with the accepted operations in mathematics, while also saying I saw interesting happenings in undefined operations. Sigh.)
To use the results of calculation with linear representations of fundamentally quadratic natural phenomena the answer cannot be "accidentally real" - nature is actually real.
Happy Trails, Len
You are more used to the math part.
Physics is not the same, in that it absorbs empirical components,
for example Schrodinger is not provable in any mathematical sense,
and almost all quantities carry units.
We need the criteria of something that actually works, not just math rigour.
Mathematicians themselvs periodically go through some rigour crisis.
i itself comes from quadratic polynomial solution, but its not really needed.
You can discuss C using (x,y) brackets plus the usual rules. Then i is merely
(0,1) in that language. Going to more complicated structure, brackets or matrices, i melts away.
In many problems it does not matter if the wave function is real or Complex. It works out the same, the wave function is equivalent by a phase factor,
that washes out with probability density.
Dear Leonard Hall, I would like to make the following comments regarding your question, even though I do not fully understand your reasoning.
Using the so called least upper bound (or supremum) property of the real numbers, which is a statement that expresses the completeness of the real numbers; i.e. the fact that every convergent real sequence converges in the real numbers; i.e. its limit is also a real number, it can be straightforwardly shown that there exists a positive real number x such that x2=2 and that that number is unique. Then x is defined to be the square root of 2. Hence, even though the square root of 2 cannot be calculated precisely, it is well defined; its definition is rigorous. Both the existence and uniqueness of the square root of 2 are derived from the completeness of R.
On the other hand, the imaginary unit i is defined to be an element of a field that contains R and in which the equation x2+1=0 has a solution; in particular i is one of the two solutions of the previous equation in the new field whose elements are the complex numbers.
Hence, the existence of i is axiomatic, contrary to the existence of the square root of 2, which is derivable from the completeness of R.
As for the probability (density) in quantum mechanics, this is just a physical interpretation of the wave function; it is an axiom of quantum mechanics; it has nothing to do with – or better, it is independent of – mathematics, since it can be neither proved nor disproved mathematically. It is justifiable in physics, since it is consistent with experimental results, but there is no guarantee that it will continue to be.
So, I think that the square root of 2, the imaginary unit, and the probability density in quantum mechanics are different and non-combinable things.
Howdy Spiros Konstantogiannis,
I appreciate your comments adding to valuable observations of other folks here. Your presentation of the rigor of the definition of the square root of 2 followed by your observation that it cannot be precisely calculated is great. I have been struggling to clarify what lies uneasy on my mind and you have stated it clearly: Hence, even though the square root of 2 cannot be calculated precisely, it is well defined; its definition is rigorous.
As an alternative contribution Juan Weisz has emphasized the nature of physics and the value therein of results. His observations are similar to your comments on the probability density function in quantum mechanics, including it has nothing to do with – or better, it is independent of – mathematics. I appear to be caught between two successful disciplines, mathematics and science - rigor and usable results - but it turns out that my curiosity thrives in just such areas for whatever that contribution is worth. I'm just the scribe, the associations are spontaneous.
"Independent of mathematics" is an interesting phrase to apply to physics. At its core the phrase is absolutely valid because natural processes do not consult theorems while happening. It follows that human linear representations also have no effect in the event, quadratic or not (of course!), they merely enable tracking it and the action required to produce a real result from the calculation activity is part of the mechanism.
However, the awareness that the irrational square root of two cannot be stated precisely and that i is also clearly different from the rationals may justify working with rational numbers, irrational numbers, and imaginary numbers as I suggested at first. That effort may produce results that are masked now. Were that the case it would be worth exploring. Sometimes a shadow is needed to see the other light that also is there.
Well, I have been certified as capable of learning in several ways and it has happened again, so perhaps we should leave it here. Thank you all.
Happy Trails, Len
Howdy Juan Weisz,
Actually, I also have read extensively in physics. I do understand that the use of i is only one way to treat C as you have clearly stated. I do not think that addresses the idea of my observation and inquiry. " A rose by any other name . . .." is the issue - my query lies behind its form of expression.
How else could I explain? Hmmmm . . .
A reply from Spiros Konstantogiannis came in at this time and my reply to him may help here. Please check that out.
Happy Trails, Len
Dear Leonard Hall, I agree with your last posts. I would like to add one more comment, regarding the phrase “independent of mathematics”. According to mathematical logic, which is the basis of mathematics, a mathematical theory is a collection of axioms, and theorems that are derivable from the axioms using rules of inference (rules of logic). Both axioms and theorems are mathematical sentences. A sentence of a theory is said to be independent of the theory if adding that sentence to the axioms of the theory yields a consistent theory AND adding the negation of that sentence to the axioms of the theory also yields a (different) consistent theory. A consistent theory is a theory whose axioms do not lead to contradictions.
In view of the previous definitions, the physical interpretation of the square of the absolute value of the wave function is independent of the mathematical theory that is used to describe quantum mechanics. The mathematical theory will remain (mathematically) valid even if the square of the absolute value of the wave function is not interpreted as probability density.
Physics: We use mathematics, with other ideas in mind. (To model something). Actually if we use axiom, it is a term a bit weaker than in math.
We could use the term eigenstate or eigevector in liu of wave function.
A vector with infinite components becoming a function or wave function.
Sometimes called state function also. In some problems H is a finite matrix
and the state is some finite eigen vector.
Howdy Folks,
I am satisfied that mathematics and physics (science) have been well defined and described here. A couple movies are running in my mind's eye that I wish to pass along as afterwords - they are observations not insults.
In Ray Bradbury's work "Medicine for Melancholy" he includes a prose movie of Pablo Picasso sketching a mural in the moist sand of a long beach as the tide is coming in - just like the "then current" theories in science presented by academics as truth, the foaming edges of the waves wash the mural away - new paradigms replace old and the human creation of science is adjusted. It is not "nature" even now.
M. C. Escher's "Metamorphosis" is a great contribution to defined elements fitted together perfectly into a closed, consistent, unnatural whole. Rigorous, however independent of nature, EXCEPT FOR THE FACT that humans and their imagination are natural. I disagree with the separation of "artificial" from natural, except as a verbal convenience.
These are creations of human minds, not figments of human imagination. And I carefully avoided an observation that fresh ideas are not fertilizer and to bury them in a field will not benefit flowers there.
Great exchange, Thanks again, Happy Trails, Len
Howdy Folks,
Perhaps I should not return to this discussion, but I think I am beginning to understand my discontent and I shall pass it along - others may feel discontent also. Remember, I seek movies in my mind that illustrate the nature of Nature. It is a good thing that formal means of expressing what little we know so far do exist, of course, and their value of rigor and use is not discounted here.
However, since nature is not mystified by our oxymorons and paradoxes, I hold out the hope we can get beyond them in our efforts to express natural processes. I raised the question whether some natural processes are essentially quadratic as part of that "quest." Were that the case, new perspectives would become available, like oxygen replacing phlogiston in our understanding of combustion.
The Schrodinger Equation, i, and the square root of two were cited in my original post and now I add treating particles of spin-1/2 as disturbing to me when expressed as a natural condition, or a rigorous representation. Should they not be treated as linear representations of essentially quadratic phenomena, fermions, which become "real" only after being "squared?" This is another illustration that attempts to express my underlying discontent with using formal modern physics and mathematics to guide movies in my mind.
Action is quantized and there is no "uncertainty" in its value, but when it is factored into energy × time or momentum × length in an individual particle representation, one finds uncertainty appears in the mathematical expressions. Wikipedia: Mathematically, in wave mechanics, the uncertainty relation between position and momentum arises because the expressions of the wavefunction in the two corresponding orthonormal bases in Hilbert space are Fourier transforms of one another (i.e., position and momentum are conjugate variables). A nonzero function and its Fourier transform cannot both be sharply localized at the same time. So, is action essentially quadratic and are its linear representations misleading? "Is" momentum or length, "is" energy or time?
I remain discontented and would like to see action in action, say as a clearly represented waveform or other visualization. [NO, these things need not be mysterious and will not remain mystifying if humanity survives long enough.]
Happy Trails, Len
Howdy Folks,
An illustration overcame efforts to fall asleep, so . . .
Purpose: to acknowledge the present effective mathematical approaches to science while suggesting a possible opening for discovery beyond them.
Say you have a conjugate pair of objects which will become clear in a moment. You may trace the gunnel of the starboard object from the bow to the stern, across to the center of the transom, down to the keel, then along the keel to the bow where you started. You traced a clockwise circuit that encloses half a vessel which may be centimeters to hundreds of meters long. You also may trace the gunnel of the larboard object from the bow to the stern, across to the center of the transom, down to the keel, then along the keel to the bow where you started. You traced a counter-clockwise circuit that also encloses half a vessel. You may engage in extensive description of these objects, create a whole zoo of objects of their types, form predictive rules and perhaps discover previously unknown designs, etc. This is valuable exploration.
If a child is drowning in a lake and you wish to rescue the child, you must glue a conjugate pair of your objects together to form a real vessel that will float to abide by the nature of Nature! My mind movie of your rescue effort is quite pleasant to watch and to comprehend.
Happy Trails, Len
Howdy Juan Weisz,
Yes, the Dirac spinor of fermions! Actually, Paul Dirac and Max Born are my favorites of that era of discovery that expanded the ideas of Albert Einstein and Louis de Broglie into numerous approaches to quantum mechanics. The treatment of pair production and Dirac's thoughts on voids in a sea of negative energy electrons to justify positrons in Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, pp. 46-47, were a delight to read.
However, the quest of this thread on ResearchGate Q&A is whether some natural processes are fundamentally quadratic, and the Schrodinger Equation, i, the square root of 2, and spin are merely examples that I hope will lead to a clear answer to that quest. My separate illustration of the conjugate objects of the starboard and larboard halves of a lifeboat that fail to support rescue of a child if they are not joined seems informative to me in this quest. These objects may be very valuable for design and for discovery of all aspects of water vessels, but they fail as boats to rescue a child individually. Boats are quadratic (?), well perhaps they are as discovery aids. (The discussion of action in the post is more significant than spin.)
Personal Note, after all each of us is actually a person: I am delighted for others that they have lives full of valuable activity so that my quests become too obscure for their available time (or, of course, are properly identified as unworthy of it). It is a regular occurrence for me, but I am happy for them! For myself, I delight in discovery like a little boy watching the fringes of waves swirl around pretty pebbles on a beach of the "ocean of truth" and I do poorly as a grown-up, but I do understand the value to others of not being distracted by youth and curiosity. (People grow up on me, or were adults already, sigh.)
From my initial post: Note: most people do not think like I do, and almost everyone is happy about that: please read openly, exploringly, as if there might be something here.
Well, maybe . . . Happy Trails, Len
Howdy Stam Nicolis , Juan Weisz , Spiros Konstantogiannis ,
Since you had been involved in this discussion months back, I have alerted you to this addition, hopefully as a courtesy. If it is an intrusion, please pardon.
HAH!! I thought there was something more in this intuition/inquiry than we had seen yet. A quote about the Dirac Operator:
It may be defined as the result of factorization of a second-order differential operator in the Minkowski space.
Reference is here on RG: Chapter - Dirac Operator and Its Properties, March 2015, DOI: 10.1007/978-3-642-40766-6_22, In book: Handbook of Relativistic Quantum Chemistry, Edition: 1, Chapter: 1, Publisher: Springer-Verlag Berlin Heidelberg 2015 Jacek Karwowski
I expect this discussion has lost its charm and that it may stop here, but I do hope that is not true. This example is a second-order differential operator which expands the set from "2," "-1," "Schrödinger Equation," "lifeboat with 'factored' starboard and larboard halves," etc., but that helps to clarify the mental unrest that raised the discussion topic.
Well, I shall read the cited article and see if it helps with my "need to know" and exposition of my disturbance. The Dirac Operator derivation, read long ago, may actually have been behind this topic unaware.
Happy Trails, Len
I dont think people think in that direction. If you removed all the quadratic or other stuff from your number system,
what will you be left with.? Well, i can indeed be considered something borrowed from another system (the complex);
you cannot either get the perfect infinitesimal out of the real. Then you need to borrow something else.
Leibniz identity, a slightly enlarged rectangle of sides x and y. The increment d(xy) is from
(x+dx)(y+dy) - xy = x dy + y dx + dx dy
but dx dy wont go away magically.
Howdy Juan Weisz,
I agree! but I'm not trying to remove quadratics, in fact, instead, I believe that I am seeking to understand treatment of phenomena that are necessarily quadratic.
Anyway, I found the following quotes in the reference pages 6 & 7 which I expect are well-known, but as we each have noted "I don't think that way." and I am learning my way with appreciated help.
"Conclusion: In a Lorentz-covariant model, the space coordinates of a particle and the time corresponding to this particle have to appear on an equal footing. In particular, if an equation is of the first-order in time, it must be of the first-order in all coordinates." and "Conclusion: In a Lorentz-covariant model, the space coordinates of a particle and the time corresponding to this particle have to appear on an equal footing. In particular, if an equation is of the first-order in time, it must be of the first-order in all coordinates."
But HoS = p2/2m, Eqn. (8), describes quantum dynamics of non-relativistic particles . . .. Thus, the energy of the system is essentially quadratic. The quoted condition on a Lorentz-covariant model requires linear representation of the quadratic term in a relativistic treatment; I have come to think that that linearization isolates the quality of the Hamiltonian/Energy in each of two relative reference systems so they may be treated individually in Lorentz-covariance. We use one of two conjugate linearized representations in some Lorentz-covariant process to calculate a result, which will be combined in an appropriate manner to restore the quadratic character of the phenomena. This appears, for the moment, to be a description of what a linearized representation of an essentially quadratic phenomena "is" in this case. The conjugate starboard - larboard halves of the lifeboat look pretty appropriate in this view.
It looks like the linearized representations are richer in content and that enables Lorentz-covariant treatment that is scraped off by the quadratic character, for instance 4 may be 22 or (-2)2, and it matters to the phenomena. In a sense, it's like the puzzles of finding a treasure when a landmark is missing, using complex numbers with a real result. The intervening steps must not be "real" because too much information from the map is lost in reality at that stage, and thereby the treasure is lost.
Well, it's all fresh thought, and I'm only on page 7 / 45, so I better read more.
Happy Trails, Len
Big switch agsin to physics
Gess you know that
H becomes linear in dirac
Aided by matrices
That are so different
Than numbers
Howdy Juan Weisz,
Yes, but how and why does: "H becomes linear in dirac, Aided by matrices That are so different Than numbers."
It now seems to me that in order to do Lorentz-covariance with a single electron, H must be written linear. In order to calculate when H is linear, matrices are necessary to carry the dual load of conjugate linear representations. Learning is interesting.
Both images in the mind and equations are dangerous. Mysteries abound if either is abused. Bully educator in High School algebra "teaching" about complex numbers with a problem: x2 + y2 = 1 and x = 2. Oh, no! big mystery, they do not intersect, solution must be imaginary! Pfffft! Use (x, y, iy) coordinate system; x = 2 is a plane, x2 + y2 = 1 is a circle in the (x, y) plane and does not intersect x = 2 plane, but x2 - (iy)2 = 1 is an hyperbola in the (x, iy) plane and that hyperbola does intersect the x = 2 plane. Some mystery, sigh.
I remain somewhat concerned about the natural state of the linearized representations, but that is a mistake on my part, because they are not natural. The linearized representations "exist" only within the calculation as carriers of information, as noted in my last post, like the intersection of the hyperbola with the plane in the complex number anecdote. Real is different.
Happy Trails, Len
Linear in algebra may mean y=ax+b.
However linear in QM means linear operators, altogether different
It means
H (a psi1 +b psi2) = a H psi1 + b H psi2
For example xx, d/dx =D , DD, (xx+ax +b), all act linearly as part of an operator.
But in fact the Dirac operator is chosen to act as in special relativity.
In free form, for free electrons, to reproduce EE=ppcc+m(0)m(0)cccc, in all in turn equivalent to E=mcc
(that is after squaring H). Look into "Dirac" matrices , the alfa(i) and beta.
Even pp/2m is a linear operator in QM
Howdy Juan Weisz,
I think there is an actual universe that is not created by our inadequate philosophies. We learn about it when we look both in and beyond them, that is, ". . . more things in Heaven and Earth . . ." applies for me. There are many issues in which I cannot get past other's confidence that "Our equations say abcd, therefore abcd." That is the "in" part and has value, but persons also use theory and equations as if they created that portion of the actual universe. As I recall, back in '69, even Encyclopedia Britannica acknowledged that rockets could function in space without air to push against; theories change and Aristotle himself would have objected to the way his learning became authority in The Schools.
I do appreciate the clarification you provide on the existing theory and approach and I appreciate your focus as you have stated it in many ways in this discussion, (in Physics )"We need the criteria of something that actually works, not just math rigour." for instance, but it is not enough for me. I want "how" and "why" answered as well as "what can we use" or "something that actually works."
Whether operators or algebra, the opportunity is to address the nature of Nature in human terms we may use to comprehend and apply. That we may abstract from actual to mathematics, operate, and make actual in the end does not explain in itself. That is why your valuable, accurate observations do not satisfy me and leave and "The Unanswered Question," as in Charles Ives musical piece. I'm afraid that shall not change.
Incidentally, "(that is after squaring H)" looks like affirmation that the result is essentially quadratic after all. Perhaps, someone who sees "it" both ways in depth will provide "the large picture" for us. Meanwhile, I shall read further.
Happy Trails, Len
Leonard (Leonardo?)
Just a short extra note. You may be interested in the so called velocity problem in Dirac theory,
described in his book, that gives way to all sorts of wrong(in my view)strange interpretations such as
Zitterbewegung or zig-zagging.
The error in reasoning here was not squaring H first and then taking a derivative with respect to momentum.
This was pointed out by Walter Greiner, in a book on QM.
Because the matrices for H(p1) need not commute with that of H(p2), so not both exist simultaneously,
in diagonal form, the derivative is invalid. At least thats my feeling, its delicate stuff. So the issue of your kind
of linearity may be improtant.
Howdy Juan Weisz,
I've been away reading "The Dirac Operator and Its Properties." It was a great review and I learned a number of things, like realizing that i and slash-h are included because one is preparing an exponent for an exponential wave equation. Radian measure was needed, but Planck had considered cycles of harmonic oscillators, thus h/2pi, and of course i is required for an exponential wave equation exponent. That the Dirac approach evolved in a few years to QED, and that Dirac was involved in its development was new, since I had gotten my QED from Feynman's lectures.
As noted in earlier posts: "I have come to think that that linearization isolates the quality of the Hamiltonian/Energy in each of two relative reference systems so they may be treated individually in Lorentz-covariance." and "It now seems to me that in order to do Lorentz-covariance with a single electron, H must be written linear. In order to calculate when H is linear, matrices are necessary to carry the . . . load of . . . linear representations." And from home notes: Incidentally, there are a large number of zeros and ones in these celebrated matrices implying that their function is to select relationships more than to modify them. and The Hamiltonian expression of energy in its linear operator persona turns out to be an expression with a complex conjugate. It may used to calculate, but the calculations produce a tentative result that must be actualized by multiplying the resulting Hamiltonian operator expression by its complex conjugate. Finally, as posted earlier: I remain somewhat concerned about the natural state of the linearized representations, but that is a mistake on my part, because they are not natural. The linearized representations "exist" only within the calculation as carriers of information . . .. They are tools, not part of the phenomena. The discussion question appears to be answered.
I shall explore the issues further, including the velocity problem in Dirac theory that you note, and I have designs on a more formal treatment, but this picture is satisfying for the moment. Thank you for your good wishes and help; I do believe I have found the sense of that for which I was looking.
Happy Trails, Len
P.S. Leonard was my father's name chosen by Norwegian immigrants to Minnesota. lfh
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