Infinities in Physical . . . How This Problem Is . . . Problem with This . . . Interval (Set) Uncertainty Let Us Try to Find a . . . A Usual Quantum- . . . as a Possible Way It Is Thus Reasonable . . . Our Hope to Avoid Infinities Acknowledgments in Physical Theories Home Page Title Page Olga Kosheleva and Vladik Kreinovich ◭◭ ◮◮ University of Texas at El Paso ◭ ◮ 500 W. University, El Paso, TX 79968, USA Page 1 of 15 olgak@utep.edu, vladik@utep.edu Go Back Full Screen Close Quit
Infinities in Physical . . . 1. Infinities in Physical Theories: a Problem How This Problem Is . . . Problem with This . . . • In many physical computations, we get meaningless Let Us Try to Find a . . . infinite values for the desired quantities. A Usual Quantum- . . . • This can be illustrated on an example of a simple prob- It Is Thus Reasonable . . . lem: Our Hope Acknowledgments – computing the overall energy of the electric field Home Page – of a charged elementary particle. Title Page • Due to relativity theory, an elementary (un-divisible) ◭◭ ◮◮ particle is a single point. ◭ ◮ • Indeed, otherwise, the particle would: Page 2 of 15 – contrary to the elementarity assumption, Go Back – consist of several not-perfectly-correlated parts. Full Screen Close Quit
Infinities in Physical . . . 2. Infinities in Physical Theories (cont-d) How This Problem Is . . . Problem with This . . . • The energy E can be obtained by integrating the en- Let Us Try to Find a . . . ergy density ρ ( x ) over the whole space: A Usual Quantum- . . . � It Is Thus Reasonable . . . E = ρ ( x ) dx. Our Hope Acknowledgments • The energy density is proportional to the square of the Home Page electric field E ( x ): Title Page ρ ( x ) ∼ E 2 ( x ) . ◭◭ ◮◮ • Due to Coulomb Law, we have E ( x ) ∼ 1 r 2 , where r is ◭ ◮ the distance to the particle’s center. Page 3 of 15 • Thus, ρ ( x ) = c r 4 . Go Back Full Screen Close Quit
Infinities in Physical . . . 3. Infinities in Physical Theories (cont-d) How This Problem Is . . . Problem with This . . . • Since this expression is spherically symmetric, we have Let Us Try to Find a . . . dx = 4 π · r 2 dr , hence A Usual Quantum- . . . � ∞ � c r 4 · 4 π · r 2 dr = It Is Thus Reasonable . . . E = ρ ( x ) dx = Our Hope 0 � ∞ ∞ � Acknowledgments r − 2 dr = r − 1 � c · = ∞ . Home Page � � 0 r =0 Title Page • This problem remains when we consider quantum equa- ◭◭ ◮◮ tions instead of classical ones. ◭ ◮ Page 4 of 15 Go Back Full Screen Close Quit
Infinities in Physical . . . 4. How This Problem Is Solved Now How This Problem Is . . . Problem with This . . . • The usual way to solve this problem is by renormaliza- Let Us Try to Find a . . . tion : crudely speaking, A Usual Quantum- . . . – we assume that the proper mass m 0 of the particle It Is Thus Reasonable . . . is m 0 = −∞ , Our Hope – then the overall energy m 0 · c 2 + E is finite. Acknowledgments Home Page • To be more precise, we consider particles of radius ε , Title Page in which case the energy E ( ε ) is large but finite. • We take a mass m 0 ( ε ) for which m 0 ( ε ) · c 2 + E ( ε ) is ◭◭ ◮◮ equal to the-determined finite value. ◭ ◮ • Then, we tend ε to 0. Page 5 of 15 Go Back Full Screen Close Quit
Infinities in Physical . . . 5. Problem with This Solution How This Problem Is . . . Problem with This . . . • From the purely computational viewpoint, renormal- Let Us Try to Find a . . . ization often works. A Usual Quantum- . . . • However, it is not a physically meaningful procedure. It Is Thus Reasonable . . . Our Hope • To many physicists, it looks more like a mathematical Acknowledgments trick. Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 6 of 15 Go Back Full Screen Close Quit
Infinities in Physical . . . 6. Let Us Try to Find a More Physically Mean- How This Problem Is . . . ingful Solution to the Problem Problem with This . . . Let Us Try to Find a . . . • One of the most important physical features of quan- A Usual Quantum- . . . tum physics is its uncertainty. It Is Thus Reasonable . . . • In the traditional non-quantum physics, equations are Our Hope deterministic. Acknowledgments Home Page • In quantum physics: Title Page – we can only measure the state of the article with uncertainty, and ◭◭ ◮◮ – we can only predict the probabilities of different ◭ ◮ future events, Page 7 of 15 – we can no longer predict the actual events with Go Back 100% guarantee. Full Screen Close Quit
Infinities in Physical . . . 7. Towards a More Physically Meaningful Solu- How This Problem Is . . . tion (cont-d) Problem with This . . . Let Us Try to Find a . . . • It is therefore reasonable to use this feature to solve A Usual Quantum- . . . the above problem. It Is Thus Reasonable . . . • We usually consider the particle located at one specific Our Hope point. Acknowledgments Home Page • Instead, let us take into account that: Title Page – due to quantum-related uncertainty, ◭◭ ◮◮ – the particle can turn out to be at different locations. ◭ ◮ Page 8 of 15 Go Back Full Screen Close Quit
Infinities in Physical . . . 8. A Usual Quantum-Physics Way of Describing How This Problem Is . . . Uncertainty Still Leads to Infinities Problem with This . . . Let Us Try to Find a . . . • Uncertainty in quantum physics is usually described by A Usual Quantum- . . . a probability distribution. It Is Thus Reasonable . . . • Let us thus try to use this approach. Our Hope Acknowledgments • For example, let us assume that the particle is Home Page – located within distance ε > 0 from the origin, Title Page – with, e.g., uniform distribution. ◭◭ ◮◮ • This distribution corresponds to equal probability ρ ( z ) = ◭ ◮ const of finding its location z anywhere within the cor- responding sphere. Page 9 of 15 c Go Back • Then, for each point x , the energy E ( x ) = | x − z | 4 Full Screen depends on z . Close Quit
Infinities in Physical . . . 9. Probabilistic Approach (cont-d) How This Problem Is . . . Problem with This . . . • So it makes sense to consider the average energy Let Us Try to Find a . . . � c A Usual Quantum- . . . | x − z | 4 · ρ ( z ) dz. It Is Thus Reasonable . . . Our Hope • However, in the vicinity z ≈ x , we have the same di- Acknowledgments vergent integral as before. Home Page • In a nutshell: Title Page – if we use probabilistic approach to describe the cor- ◭◭ ◮◮ responding uncertainty, ◭ ◮ – the problem gets only worse. Page 10 of 15 • At least, before we had a finite value for the energy Go Back density. Full Screen • Now even energy density itself is infinite. Close Quit
Infinities in Physical . . . 10. Probabilistic Approach (cont-d) How This Problem Is . . . Problem with This . . . • We have shown that the value is infinite for the uniform Let Us Try to Find a . . . distribution. A Usual Quantum- . . . • However, one can show that it is infinite for any distri- It Is Thus Reasonable . . . bution with ρ ( z ) > 0 for some z . Our Hope Acknowledgments Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 11 of 15 Go Back Full Screen Close Quit
Infinities in Physical . . . 11. It Is Thus Reasonable to Consider Interval How This Problem Is . . . (Set) Uncertainty Problem with This . . . Let Us Try to Find a . . . • We do not know the exact location of a particle. A Usual Quantum- . . . • So it is not reasonable to assume that we know the It Is Thus Reasonable . . . exact probabilities either. Our Hope Acknowledgments • The simplest case is: Home Page – when we have no information about the correspond- Title Page ing probabilities, ◭◭ ◮◮ – when all we know is, e.g., that the particle is located within the sphere of radius ε . ◭ ◮ • Since we do not know probabilities, we cannot compute Page 12 of 15 the average density. Go Back • We can only compute, for each point x , the smallest Full Screen and largest possible values of the energy density. Close Quit
Infinities in Physical . . . 12. Interval (Set) Uncertainty (cont-d) How This Problem Is . . . Problem with This . . . • For a point at distance r from the center: Let Us Try to Find a . . . – the smallest possible value of the energy density is A Usual Quantum- . . . – when the particle is the farthest away from this It Is Thus Reasonable . . . point, at the distance r + ε . Our Hope Acknowledgments • The largest possible value is when the particle is the Home Page closest, at distance max( r − ε, 0). Title Page c • For the largest density ρ ( x ) = (max( r − ε, 0)) 4 , we still ◭◭ ◮◮ get infinite overall energy. ◭ ◮ c • However, for the smallest energy density ρ ( x ) = ( r + ε ) 4 , Page 13 of 15 we get a finite overall energy. Go Back • And we get this finite value no matter what is the shape Full Screen of the region in which the particle is located. Close Quit
Infinities in Physical . . . 13. Our Hope How This Problem Is . . . Problem with This . . . • This example makes us believe that: Let Us Try to Find a . . . – such an interval (set) uncertainty A Usual Quantum- . . . – can help avoid infinities in other situations as well. It Is Thus Reasonable . . . Our Hope Acknowledgments Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 14 of 15 Go Back Full Screen Close Quit
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