Why Feynman Feynmans Approach: . . . First Idea: An . . . Second - PowerPoint PPT Presentation

Why Use Quantum . . . Need for Quantization Why Feynman Feynman’s Approach: . . . First Idea: An . . . Second Idea: . . . Path Integration? Third Idea: Maximal . . . Fourth Idea: . . . Jaime Nava 1 , Juan Ferret 2 , and Vladik Kreinovich 1 Title Page 1 Department of Computer Science ◭◭ ◮◮ 2 Department of Philosophy ◭ ◮ University of Texas at El Paso 500 W. University Page 1 of 15 El Paso, TX 79968, USA jenava@miners.utep.edu Go Back jferret@utep.edu Full Screen vladik@utep.edu Close Quit

1. Why Use Quantum Effects in Computing? Why Use Quantum . . . Need for Quantization • Fact: computers are fast. Feynman’s Approach: . . . • Challenge: computer are not yet fast enough: First Idea: An . . . Second Idea: . . . – to predict weather more accurately and earlier, Third Idea: Maximal . . . – to run industrial robots more efficiently, Fourth Idea: . . . – to power complex simulations of forest fires, Title Page and for many other applications, we need higher com- ◭◭ ◮◮ puting speed. ◭ ◮ • How: to increase the computing speed, we must make signals travel faster between computer components. Page 2 of 15 Go Back • Limitation: signals cannot travel faster than the speed of light, and they already travel at about that speed. Full Screen • Conclusion: the size of computer components must be Close reduced. Quit

2. Why Quantum Effects in Computing? (cont-d) Why Use Quantum . . . Need for Quantization • Reminder: the size of computer components must be Feynman’s Approach: . . . reduced. First Idea: An . . . • Fact: the sizes of these components have almost reached Second Idea: . . . the sizes of atoms and molecules. Third Idea: Maximal . . . Fourth Idea: . . . • Fact: these molecules follow the rules of quantum me- chanics. Title Page • Conclusion: quantum computing is necessary. ◭◭ ◮◮ • Resulting question: how to describe these quantum ef- ◭ ◮ fects? Page 3 of 15 Go Back Full Screen Close Quit

3. Need for Quantization Why Use Quantum . . . Need for Quantization • Since the early 1900s, we know that we need to take Feynman’s Approach: . . . into account quantum effects. Thus: First Idea: An . . . – for every non-quantum physical theory describing Second Idea: . . . a certain phenomenon, be it Third Idea: Maximal . . . ∗ mechanics Fourth Idea: . . . ∗ or electrodynamics Title Page ∗ or gravitation theory, ◭◭ ◮◮ – we must come up with an appropriate quantum the- ◭ ◮ ory. Page 4 of 15 • Traditional quantization methods: replace the scalar physical quantities with the corresponding operators. Go Back • Problem: operators are non-commutative, px � = xp . Full Screen Close • Conclusion: several different quantum versions of each classical theory. Quit

4. Towards Feynman’s Approach: Least Action Prin- Why Use Quantum . . . ciple Need for Quantization Feynman’s Approach: . . . • Laws of physics have been traditionally described in First Idea: An . . . terms of differential equations. Second Idea: . . . • For fundamental physical phenomena, not all differen- Third Idea: Maximal . . . tial equations make sense. Fourth Idea: . . . • Example: we need conservation of fundamental physi- Title Page cal quantities (energy, momentum, etc.) ◭◭ ◮◮ • It turns out that ◭ ◮ – all known fundamental physical equations can be Page 5 of 15 described in terms of minimization, and Go Back – in general, equations following from a minimization Full Screen principle lead to conservation laws. Close Quit

5. Towards Feynman’s Approach: Least Action Prin- Why Use Quantum . . . ciple (cont-d) Need for Quantization Feynman’s Approach: . . . • Idea: we can assign, to each trajectory γ ( t ), we can First Idea: An . . . assign a value S ( γ ) such that Second Idea: . . . – among all possible trajectories, Third Idea: Maximal . . . – the actual one is the one for which the value S ( γ ) Fourth Idea: . . . is the smallest possible. Title Page • This value S ( γ ) is called action . ◭◭ ◮◮ • The principle that action is minimized along the actual ◭ ◮ trajectory is called the minimal action principle . Page 6 of 15 • Feynman’s idea: the probability to get from the state Go Back γ to the state γ is proportional to | ψ ( γ → γ ) | 2 , where Full Screen � � i · S ( γ ) � ψ = exp . Close � γ : γ → γ Quit

6. Feynman’s Approach: Successes and Challenges Why Use Quantum . . . Need for Quantization • Successes: Feynman’s approach is an efficient comput- Feynman’s Approach: . . . ing tool: First Idea: An . . . – we can expand the corresponding expression, and Second Idea: . . . Third Idea: Maximal . . . – represent the resulting probability as a sum of an infinite series. Fourth Idea: . . . • Each term of this series can be described by an appro- Title Page priate graph called Feynman diagram . ◭◭ ◮◮ • Foundational challenge: why the above formula? ◭ ◮ • What we do in this talk: we provide a natural expla- Page 7 of 15 nation for Feynman’s path integration formula. Go Back Full Screen Close Quit

7. First Idea: An Alternative Representation of the Why Use Quantum . . . Original Theory Need for Quantization Feynman’s Approach: . . . • Reminder: a physical theory is a functional S that First Idea: An . . . assigns, to every path γ , the value of the action S ( γ ). Second Idea: . . . • From this viewpoint, a priori, all the paths are equiva- Third Idea: Maximal . . . lent, they only differ by the corresponding values S ( γ ). Fourth Idea: . . . • In other words, what is important is the frequency with Title Page which we encounter different values S ( γ ): ◭◭ ◮◮ – if among N paths, only one has this value of the ◭ ◮ action, this frequency is 1 /N , Page 8 of 15 – if two, the frequency is 2 /N , etc. Go Back • In mathematical terms, this means that we consider the action S ( γ ) as a random variable . Full Screen • One possible way to describe a random variable α is Close def by its characteristic function χ α ( ω ) = E [exp(i · ω · α )] . Quit

8. An Alternative Representation of the Original The- Why Use Quantum . . . ory (cont-d) Need for Quantization Feynman’s Approach: . . . • Reminder: we consider α = S ( γ ) as a random variable. First Idea: An . . . • Reminder: we describe α via its characteristics func- Second Idea: . . . def tion χ α ( ω ) = E [exp(i · ω · α )] . Third Idea: Maximal . . . Fourth Idea: . . . • Conclusion: χ ( ω ) = 1 � Title Page N · exp(i · S ( γ ) · ω ) . γ ◭◭ ◮◮ • Reminder: Feynman’s formula ◭ ◮ � � i · S ( γ ) Page 9 of 15 � ψ = exp . � γ : γ → γ Go Back Full Screen • Observation: Feynman’s formula is χ (1 / � ). Close • Comment: this is not yet a derivation, since there are many ways to represent a random variable. Quit

9. Second Idea: Appropriate Behavior for Indepen- Why Use Quantum . . . dent Physical Systems Need for Quantization Feynman’s Approach: . . . • Objective: derive a formula that transforms a func- First Idea: An . . . tional S ( γ ) into transition probabilities. Second Idea: . . . • Typical situation: the physical system consists of two Third Idea: Maximal . . . subsystems. Fourth Idea: . . . • In this case, each state γ of the composite system is a Title Page pair γ = ( γ 1 , γ 2 ) consisting of ◭◭ ◮◮ – a state γ 1 of the first subsystem and ◭ ◮ – the state γ 2 of the second subsystem. Page 10 of 15 • Often, these subsystems are independent . Go Back • Due to this independence, Full Screen P (( γ 1 , γ 2 ) → ( γ ′ 1 , γ ′ 2 )) = P 1 ( γ 1 → γ ′ 1 ) · P 2 ( γ 2 → γ ′ 2 ) . Close Quit

10. Independent Physical Systems (cont-d) Why Use Quantum . . . Need for Quantization • Reminder: Feynman’s Approach: . . . P (( γ 1 , γ 2 ) → ( γ ′ 1 , γ ′ 2 )) = P 1 ( γ 1 → γ ′ 1 ) · P 2 ( γ 2 → γ ′ First Idea: An . . . 2 ) . Second Idea: . . . • In physics, independence is usually described as Third Idea: Maximal . . . Fourth Idea: . . . S (( γ 1 , γ 2 )) = S 1 ( γ 1 ) + S 2 ( γ 2 ) . Title Page • In probabilistic terms, this means that we have the sum of two independent random variables. So: ◭◭ ◮◮ – the probability corresponding to the sum of inde- ◭ ◮ pendent random variables Page 11 of 15 – is equal to the product of corresponding probabili- Go Back ties. Full Screen • Fact: for the sum, characteristic functions multiply: Close χ α 1 + α 2 ( ω ) = χ α 1 ( ω ) · χ α 2 ( ω ) . Quit