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129-442: The Feynman–Kac formula , named after Richard Feynman and Mark Kac , establishes a link between parabolic partial differential equations and stochastic processes . In 1947, when Kac and Feynman were both faculty members at Cornell University , Kac attended a presentation of Feynman's and remarked that the two of them were working on the same thing from different directions. The Feynman–Kac formula resulted, which proves rigorously

258-564: A Wiener process (also called Brownian motion ) under Q {\displaystyle Q} . Suppose that the position X t {\displaystyle X_{t}} of a particle evolves according to the diffusion process d X t = μ ( X t , t ) d t + σ ( X t , t ) d W t Q . {\displaystyle dX_{t}=\mu (X_{t},t)\,\mathrm {d} t+\sigma (X_{t},t)\,\mathrm {d} W_{t}^{Q}.} Let

387-403: A random vector . The conditional expectation e X : R n → R {\displaystyle e_{X}:\mathbb {R} ^{n}\to \mathbb {R} } is a measurable function such that Note that unlike μ X {\displaystyle \mu _{X}} , the conditional expectation e X {\displaystyle e_{X}}

516-424: A "bum". The young Feynman was heavily influenced by his father, who encouraged him to ask questions to challenge orthodox thinking, and who was always ready to teach Feynman something new. From his mother, he gained the sense of humor that he had throughout his life. As a child, he had a talent for engineering, maintained an experimental laboratory in his home, and delighted in repairing radios. This radio repairing

645-515: A European call option on an S t {\displaystyle S_{t}} expiring at time T {\displaystyle T} with strike K {\displaystyle K} . At expiry, it is worth ( X T − K ) + . {\displaystyle (X_{T}-K)^{+}.} Then, the risk-neutral price of the option, at time t {\displaystyle t} and stock price x {\displaystyle x} ,

774-488: A bunch of professional soldiers," Wilson later recalled, "we signed up, en masse, to go to Los Alamos." Oppenheimer recruited many young physicists, including Feynman, who he telephoned long distance from Chicago to inform that he had found a Presbyterian sanatorium in Albuquerque, New Mexico for Arline. They were among the first to depart for New Mexico, leaving on a train on March 28, 1943. The railroad supplied Arline with

903-586: A car from his friend Klaus Fuchs to drive to Albuquerque to see Arline. Asked who at Los Alamos was most likely to be a spy, Fuchs mentioned Feynman's safe-cracking and frequent trips to Albuquerque; Fuchs himself later confessed to spying for the Soviet Union . The FBI would compile a bulky file on Feynman, particularly in view of Feynman's Q clearance . Informed that Arline was dying, Feynman drove to Albuquerque and sat with her for hours until she died on June 16, 1945. He then immersed himself in work on

1032-504: A complement to the quark model developed by Gell-Mann. The relationship between the two models was murky; Gell-Mann referred to Feynman's partons derisively as "put-ons". In the mid-1960s, physicists believed that quarks were just a bookkeeping device for symmetry numbers, not real particles; the statistics of the omega-minus particle , if it were interpreted as three identical strange quarks bound together, seemed impossible if quarks were real. Conditional expectation Depending on

1161-556: A father of the Xbox , bought the van. Qantum was the license plate ID. At Caltech, Feynman investigated the physics of the superfluidity of supercooled liquid helium , where helium seems to display a complete lack of viscosity when flowing. Feynman provided a quantum-mechanical explanation for the Soviet physicist Lev Landau 's theory of superfluidity. Applying the Schrödinger equation to

1290-425: A method for evaluating functional integrals of a certain form. If I = ∫ f ( x ( 0 ) ) exp ⁡ ( − u ∫ 0 t V ( x ( t ) ) d t ) g ( x ( t ) ) D x {\displaystyle I=\int f(x(0))\exp \left(-u\int _{0}^{t}V(x(t))\,dt\right)g(x(t))\,Dx} where

1419-658: A point of dating other men, but Feynman proposed in early 1960. They were married on September 24, 1960, at the Huntington Hotel in Pasadena. They had a son, Carl, in 1962, and adopted a daughter, Michelle, in 1968. Besides their home in Altadena, they had a beach house in Baja California, purchased with the money from Feynman's Nobel Prize. Feynman tried marijuana and ketamine at John Lilly 's sensory deprivation tanks , as

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1548-490: A probability space, and X : Ω → R {\displaystyle X:\Omega \to \mathbb {R} } in L 2 {\displaystyle L^{2}} with mean μ X {\displaystyle \mu _{X}} and variance σ X 2 {\displaystyle \sigma _{X}^{2}} . The expectation μ X {\displaystyle \mu _{X}} minimizes

1677-414: A relativistic version. Feynman thought that he could do this, but when he went back to Bethe with his solution, it did not converge. Feynman carefully worked through the problem again, applying the path integral formulation that he had used in his thesis. Like Bethe, he made the integral finite by applying a cut-off term. The result corresponded to Bethe's version. Feynman presented his work to his peers at

1806-554: A seat on the President's Science Advisory Committee , but was not appointed. At this time, the FBI interviewed a woman close to Feynman, possibly his ex-wife Bell, who sent a written statement to J. Edgar Hoover on August 8, 1958: I do not know—but I believe that Richard Feynman is either a Communist or very strongly pro-Communist—and as such is a very definite security risk. This man is, in my opinion, an extremely complex and dangerous person,

1935-472: A very dangerous person to have in a position of public trust ... In matters of intrigue Richard Feynman is, I believe immensely clever—indeed a genius—and he is, I further believe, completely ruthless, unhampered by morals, ethics, or religion—and will stop at absolutely nothing to achieve his ends. The U.S. government nevertheless sent Feynman to Geneva for the September 1958 Atoms for Peace Conference. On

2064-653: A way of studying consciousness . He gave up alcohol when he began to show vague, early signs of alcoholism , as he did not want to consume anything that could damage his brain. Despite his curiosity about hallucinations , he was reluctant to experiment with LSD . Feynman had synesthesia , and said that mathematical symbols had different colors for him: "When I see equations, I see the letters in colors. I don't know why. I see vague pictures of Bessel functions with light-tan j's, slightly violet-bluish n's, and dark brown x's flying around." There had been protests over his alleged sexism in 1968, and again in 1972. Although there

2193-421: A weather station on every day of the ten–year (3652-day) period from January 1, 1990, to December 31, 1999. The unconditional expectation of rainfall for an unspecified day is the average of the rainfall amounts for those 3652 days. The conditional expectation of rainfall for an otherwise unspecified day known to be (conditional on being) in the month of March, is the average of daily rainfall over all 310 days of

2322-476: A wheelchair, and Feynman paid extra for a private room for her. There they spent their wedding anniversary. At Los Alamos, Feynman was assigned to Hans Bethe's Theoretical (T) Division, and impressed Bethe enough to be made a group leader. He and Bethe developed the Bethe–Feynman formula for calculating the yield of a fission bomb , which built upon previous work by Robert Serber . As a junior physicist, he

2451-417: Is u ( x , t ) = E [ e − ∫ t T r s d s ( S T − K ) + | ln ⁡ S t = ln ⁡ x ] . {\displaystyle u(x,t)=E\left[e^{-\int _{t}^{T}r_{s}ds}(S_{T}-K)^{+}|\ln S_{t}=\ln x\right].} Plugging into

2580-621: Is a finite measure on ( Ω , F ) {\displaystyle (\Omega ,{\mathcal {F}})} that is absolutely continuous with respect to P {\displaystyle P} . If h {\displaystyle h} is the natural injection from H {\displaystyle {\mathcal {H}}} to F {\displaystyle {\mathcal {F}}} , then μ X ∘ h = μ X | H {\displaystyle \mu ^{X}\circ h=\mu ^{X}|_{\mathcal {H}}}

2709-403: Is absolutely continuous with respect to P ∘ h {\displaystyle P\circ h} , because the condition implies Thus, we have where the derivatives are Radon–Nikodym derivatives of measures. Consider, in addition to the above, The conditional expectation of X given Y is defined by applying the above construction on the σ-algebra generated by Y : By

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2838-426: Is an Itô process satisfying d X t = μ ( X t , t ) d t + σ ( X t , t ) d W t Q , {\displaystyle \mathrm {d} X_{t}=\mu (X_{t},t)\,\mathrm {d} t+\sigma (X_{t},t)\,\mathrm {d} W_{t}^{Q},} and W t Q {\displaystyle W_{t}^{Q}}

2967-3016: Is as follows: Let u ( x , t ) {\displaystyle u(x,t)} be the solution to the above partial differential equation. Applying the product rule for Itô processes to the process Y ( s ) = exp ⁡ ( − ∫ t s V ( X τ , τ ) d τ ) u ( X s , s ) + ∫ t s exp ⁡ ( − ∫ t r V ( X τ , τ ) d τ ) f ( X r , r ) d r {\displaystyle Y(s)=\exp \left(-\int _{t}^{s}V(X_{\tau },\tau )\,d\tau \right)u(X_{s},s)+\int _{t}^{s}\exp \left(-\int _{t}^{r}V(X_{\tau },\tau )\,d\tau \right)f(X_{r},r)\,dr} one gets: d Y s = d ( exp ⁡ ( − ∫ t s V ( X τ , τ ) d τ ) ) u ( X s , s ) + exp ⁡ ( − ∫ t s V ( X τ , τ ) d τ ) d u ( X s , s ) + d ( exp ⁡ ( − ∫ t s V ( X τ , τ ) d τ ) ) d u ( X s , s ) + d ( ∫ t s exp ⁡ ( − ∫ t r V ( X τ , τ ) d τ ) f ( X r , r ) d r ) {\displaystyle {\begin{aligned}dY_{s}={}&d\left(\exp \left(-\int _{t}^{s}V(X_{\tau },\tau )\,d\tau \right)\right)u(X_{s},s)+\exp \left(-\int _{t}^{s}V(X_{\tau },\tau )\,d\tau \right)\,du(X_{s},s)\\[6pt]&{}+d\left(\exp \left(-\int _{t}^{s}V(X_{\tau },\tau )\,d\tau \right)\right)du(X_{s},s)+d\left(\int _{t}^{s}\exp \left(-\int _{t}^{r}V(X_{\tau },\tau )\,d\tau \right)f(X_{r},r)\,dr\right)\end{aligned}}} Since d ( exp ⁡ ( − ∫ t s V ( X τ , τ ) d τ ) ) = − V ( X s , s ) exp ⁡ ( − ∫ t s V ( X τ , τ ) d τ ) d s , {\displaystyle d\left(\exp \left(-\int _{t}^{s}V(X_{\tau },\tau )\,d\tau \right)\right)=-V(X_{s},s)\exp \left(-\int _{t}^{s}V(X_{\tau },\tau )\,d\tau \right)\,ds,}

3096-560: Is developed without this assumption, see below under Conditional expectation with respect to a sub-σ-algebra . The L 2 {\displaystyle L^{2}} theory is, however, considered more intuitive and admits important generalizations . In the context of L 2 {\displaystyle L^{2}} random variables, conditional expectation is also called regression . In what follows let ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},P)} be

3225-562: Is equivalent to linear regression: for coefficients { α i } i = 0.. n {\displaystyle \{\alpha _{i}\}_{i=0..n}} described in Multivariate normal distribution#Conditional distributions . Consider the following: Since H {\displaystyle {\mathcal {H}}} is a sub σ {\displaystyle \sigma } -algebra of F {\displaystyle {\mathcal {F}}} ,

3354-592: Is introduced with the meaning E ( X ∣ Y ) = f ( Y ) {\displaystyle E(X\mid Y)=f(Y)} . Consider the roll of a fair die and let A = 1 if the number is even (i.e., 2, 4, or 6) and A = 0 otherwise. Furthermore, let B = 1 if the number is prime (i.e., 2, 3, or 5) and B = 0 otherwise. The unconditional expectation of A is E [ A ] = ( 0 + 1 + 0 + 1 + 0 + 1 ) / 6 = 1 / 2 {\displaystyle E[A]=(0+1+0+1+0+1)/6=1/2} , but

3483-661: Is known as the Hellmann–Feynman theorem . In 1939, Feynman received a bachelor's degree and was named a Putnam Fellow . He attained a perfect score on the graduate school entrance exams to Princeton University in physics—an unprecedented feat—and an outstanding score in mathematics, but did poorly on the history and English portions. The head of the physics department there, Henry D. Smyth , had another concern, writing to Philip M. Morse to ask: "Is Feynman Jewish? We have no definite rule against Jews but have to keep their proportion in our department reasonably small because of

3612-621: Is no evidence he supported discrimination against women in science, protestors "objected to his use of sexist stories about 'lady drivers' and clueless women in his lectures." Feynman recalled protesters entering a hall and picketing a lecture he was about to make in San Francisco, calling him a "sexist pig". Seeing the protesters, as Feynman later recalled the incident, he addressed institutional sexism by saying that "women do indeed suffer prejudice and discrimination in physics, and your presence here today serves to remind us of these difficulties and

3741-567: Is non-trivial. It can be shown that is a closed subspace of the Hilbert space L 2 ( Ω ) {\displaystyle L^{2}(\Omega )} . By the Hilbert projection theorem , the necessary and sufficient condition for e X {\displaystyle e_{X}} to be a minimizer is that for all f ( Y ) {\displaystyle f(Y)} in M we have In words, this equation says that

3870-906: Is not generally unique: there may be multiple minimizers of the mean squared error. Example 1 : Consider the case where Y is the constant random variable that's always 1. Then the mean squared error is minimized by any function of the form Example 2 : Consider the case where Y is the 2-dimensional random vector ( X , 2 X ) {\displaystyle (X,2X)} . Then clearly but in terms of functions it can be expressed as e X ( y 1 , y 2 ) = 3 y 1 − y 2 {\displaystyle e_{X}(y_{1},y_{2})=3y_{1}-y_{2}} or e X ′ ( y 1 , y 2 ) = y 2 − y 1 {\displaystyle e'_{X}(y_{1},y_{2})=y_{2}-y_{1}} or infinitely many other ways. In

3999-487: Is not to be expected according to Vallarta and Feynman". When they next met, Feynman asked gleefully whether Vallarta had seen Heisenberg's book. Vallarta knew why Feynman was grinning. "Yes," he replied. "You're the last word in cosmic rays." The other was his senior thesis, on "Forces in Molecules", based on a topic assigned by John C. Slater , who was sufficiently impressed by the paper to have it published. Its main result

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4128-562: Is required to be a simple function , linear regression when g is required to be affine , etc. These generalizations of conditional expectation come at the cost of many of its properties no longer holding. For example, let M be the space of all linear functions of Y and let E M {\displaystyle {\mathcal {E}}_{M}} denote this generalized conditional expectation/ L 2 {\displaystyle L^{2}} projection. If M {\displaystyle M} does not contain

4257-393: Is taken over all possible outcomes of X . If P ( A ) = 0 {\displaystyle P(A)=0} , the conditional expectation is undefined due to the division by zero. If X and Y are discrete random variables , the conditional expectation of X given Y is where P ( X = x , Y = y ) {\displaystyle P(X=x,Y=y)}

4386-1201: Is the joint probability mass function of X and Y . The sum is taken over all possible outcomes of X . Remark that as above the expression is undefined if P ( Y = y ) = 0 {\displaystyle P(Y=y)=0} . Conditioning on a discrete random variable is the same as conditioning on the corresponding event: where A is the set { Y = y } {\displaystyle \{Y=y\}} . Let X {\displaystyle X} and Y {\displaystyle Y} be continuous random variables with joint density f X , Y ( x , y ) , {\displaystyle f_{X,Y}(x,y),} Y {\displaystyle Y} 's density f Y ( y ) , {\displaystyle f_{Y}(y),} and conditional density f X | Y ( x | y ) = f X , Y ( x , y ) f Y ( y ) {\displaystyle \textstyle f_{X|Y}(x|y)={\frac {f_{X,Y}(x,y)}{f_{Y}(y)}}} of X {\displaystyle X} given

4515-417: Is the expected cost-to-go, if the particle starts at ( t , X t = x ) . {\displaystyle (t,X_{t}=x).} A proof that the above formula is a solution of the differential equation is long, difficult and not presented here. It is however reasonably straightforward to show that, if a solution exists , it must have the above form. The proof of that lesser result

4644-538: Is the restriction of μ X {\displaystyle \mu ^{X}} to H {\displaystyle {\mathcal {H}}} and P ∘ h = P | H {\displaystyle P\circ h=P|_{\mathcal {H}}} is the restriction of P {\displaystyle P} to H {\displaystyle {\mathcal {H}}} . Furthermore, μ X ∘ h {\displaystyle \mu ^{X}\circ h}

4773-473: Is the restriction of P {\displaystyle P} to H {\displaystyle {\mathcal {H}}} , cannot be stated in general. However, the local averages ∫ H X d P {\textstyle \int _{H}X\,dP} can be recovered in ( Ω , H , P | H ) {\displaystyle (\Omega ,{\mathcal {H}},P|_{\mathcal {H}})} with

4902-519: Is the risk-free interest rate and σ t {\displaystyle \sigma _{t}} is the volatility. Equivalently, by Itô's lemma, d ln ⁡ S t = ( r t − 1 2 σ t 2 ) d t + σ t d W t . {\displaystyle d\ln S_{t}=\left(r_{t}-{\tfrac {1}{2}}\sigma _{t}^{2}\right)dt+\sigma _{t}\,dW_{t}.} Now consider

5031-1143: Is the unknown. Then the Feynman–Kac formula expresses u ( x , t ) {\displaystyle u(x,t)} as a conditional expectation under the probability measure Q {\displaystyle Q} u ( x , t ) = E Q [ e − ∫ t T V ( X τ , τ ) d τ ψ ( X T ) + ∫ t T e − ∫ t τ V ( X s , s ) d s f ( X τ , τ ) d τ | X t = x ] {\displaystyle u(x,t)=E^{Q}\left[e^{-\int _{t}^{T}V(X_{\tau },\tau )\,\mathrm {d} \tau }\psi (X_{T})+\int _{t}^{T}e^{-\int _{t}^{\tau }V(X_{s},s)\,\mathrm {d} s}f(X_{\tau },\tau )\,\mathrm {d} \tau \,{\Bigg |}\,X_{t}=x\right]} where X {\displaystyle X}

5160-869: The L 2 {\displaystyle L^{2}} discussion, this condition is equivalent to saying that the residual X − E ⁡ ( X ∣ H ) {\displaystyle X-\operatorname {E} (X\mid {\mathcal {H}})} is orthogonal to the indicator functions 1 H {\displaystyle 1_{H}} : The existence of E ⁡ ( X ∣ H ) {\displaystyle \operatorname {E} (X\mid {\mathcal {H}})} can be established by noting that μ X : F ↦ ∫ F X d P {\textstyle \mu ^{X}\colon F\mapsto \int _{F}X\,\mathrm {d} P} for F ∈ F {\displaystyle F\in {\mathcal {F}}}

5289-482: The frigideira , a metal percussion instrument based on a frying pan. He was an enthusiastic amateur player of bongo and conga drums and often played them in the pit orchestra in musicals. He spent time in Rio with his friend Bohm, but Bohm could not convince Feynman to investigate Bohm's ideas on physics. Feynman did not return to Cornell. Bacher, who had been instrumental in bringing Feynman to Cornell, had lured him to

Feynman–Kac formula - Misplaced Pages Continue

5418-508: The Physical Review . One of these, which was co-written with Manuel Vallarta , was entitled "The Scattering of Cosmic Rays by the Stars of a Galaxy". Vallarta let his student in on a secret of mentor-protégé publishing: the senior scientist's name comes first. Feynman had his revenge a few years later, when Heisenberg concluded an entire book on cosmic rays with the phrase: "such an effect

5547-457: The American option do not have a fixed expiry. Some options have value at expiry determined by the past stock prices . For example, an average option has a payoff that is not determined by the underlying price at expiry but by the average underlying price over some predetermined period of time. For these, the Feynman–Kac formula does not directly apply. In quantum chemistry , it is used to solve

5676-567: The Arista Honor Society , in his last year in high school he won the New York University Math Championship. His habit of direct characterization sometimes rattled more conventional thinkers; for example, one of his questions, when learning feline anatomy , was "Do you have a map of the cat?" (referring to an anatomical chart). Feynman applied to Columbia University but was not accepted because of its quota for

5805-486: The Doob-Dynkin lemma , there exists a function e X : U → R n {\displaystyle e_{X}\colon U\to \mathbb {R} ^{n}} such that For a Borel subset B in B ( R n ) {\displaystyle {\mathcal {B}}(\mathbb {R} ^{n})} , one can consider the collection of random variables It can be shown that they form

5934-533: The Pocono Conference in 1948. It did not go well. Julian Schwinger gave a long presentation of his work in quantum electrodynamics, and Feynman then offered his version, entitled "Alternative Formulation of Quantum Electrodynamics". The unfamiliar Feynman diagrams , used for the first time, puzzled the audience. Feynman failed to get his point across, and Paul Dirac , Edward Teller and Niels Bohr all raised objections. To Freeman Dyson , one thing at least

6063-457: The Radon–Nikodym theorem . In works of Paul Halmos and Joseph L. Doob from 1953, conditional expectation was generalized to its modern definition using sub-σ-algebras . If A is an event in F {\displaystyle {\mathcal {F}}} with nonzero probability, and X is a discrete random variable , the conditional expectation of X given A is where the sum

6192-603: The Schrödinger equation with the Pure Diffusion Monte Carlo method. Richard Feynman Richard Phillips Feynman ( / ˈ f aɪ n m ə n / ; May 11, 1918 – February 15, 1988) was an American theoretical physicist . He is best known for his work in the path integral formulation of quantum mechanics , the theory of quantum electrodynamics , the physics of the superfluidity of supercooled liquid helium , and in particle physics , for which he proposed

6321-637: The University of Wisconsin–Madison as an assistant professor of physics, but was on unpaid leave during his involvement in the Manhattan Project. In 1945, he received a letter from Dean Mark Ingraham of the College of Letters and Science requesting his return to the university to teach in the coming academic year. His appointment was not extended when he did not commit to returning. In a talk given there several years later, Feynman quipped, "It's great to be back at

6450-415: The constant functions , the tower property E ⁡ ( E M ( X ) ) = E ⁡ ( X ) {\displaystyle \operatorname {E} ({\mathcal {E}}_{M}(X))=\operatorname {E} (X)} will not hold. An important special case is when X and Y are jointly normally distributed. In this case it can be shown that the conditional expectation

6579-516: The ferry to Staten Island , where they were married in the city office. The ceremony was attended by neither family nor friends and was witnessed by a pair of strangers. Feynman could kiss Arline only on the cheek. After the ceremony he took her to Deborah Hospital , where he visited her on weekends. In 1941, with World War II occurring in Europe but the United States not yet at war, Feynman spent

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6708-441: The mean squared error : The conditional expectation of X is defined analogously, except instead of a single number μ X {\displaystyle \mu _{X}} , the result will be a function e X ( y ) {\displaystyle e_{X}(y)} . Let Y : Ω → R n {\displaystyle Y:\Omega \to \mathbb {R} ^{n}} be

6837-545: The parton model . For his contributions to the development of quantum electrodynamics, Feynman received the Nobel Prize in Physics in 1965 jointly with Julian Schwinger and Shin'ichirō Tomonaga . Feynman developed a widely used pictorial representation scheme for the mathematical expressions describing the behavior of subatomic particles , which later became known as Feynman diagrams . During his lifetime, Feynman became one of

6966-579: The principle of stationary action to problems of quantum mechanics, inspired by a desire to quantize the Wheeler–Feynman absorber theory of electrodynamics, and laid the groundwork for the path integral formulation and Feynman diagrams. A key insight was that positrons behaved like electrons moving backwards in time. James Gleick wrote: This was Richard Feynman nearing the crest of his powers. At twenty-three ... there may now have been no physicist on earth who could match his exuberant command over

7095-450: The residual X − e X ( Y ) {\displaystyle X-e_{X}(Y)} is orthogonal to the space M of all functions of Y . This orthogonality condition, applied to the indicator functions f ( Y ) = 1 Y ∈ H {\displaystyle f(Y)=1_{Y\in H}} , is used below to extend conditional expectation to

7224-431: The synagogue every Friday. However, by his youth, Feynman described himself as an "avowed atheist ". Many years later, in a letter to Tina Levitan , declining a request for information for her book on Jewish Nobel Prize winners, he stated, "To select, for approbation the peculiar elements that come from some supposedly Jewish heredity is to open the door to all kinds of nonsense on racial theory", adding, "at thirteen I

7353-574: The "Mathematical Formulation of the Quantum Theory of Electromagnetic Interaction" in 1950 and "An Operator Calculus Having Applications in Quantum Electrodynamics" in 1951, he developed the mathematical basis of his ideas, derived familiar formulae and advanced new ones. While papers by others initially cited Schwinger, papers citing Feynman and employing Feynman diagrams appeared in 1950, and soon became prevalent. Students learned and used

7482-426: The "diagonal" { y : y 2 = 2 y 1 } {\displaystyle \{y:y_{2}=2y_{1}\}} , so that any set not intersecting it has measure 0. The existence of a minimizer for min g E ⁡ ( ( X − g ( Y ) ) 2 ) {\displaystyle \min _{g}\operatorname {E} \left((X-g(Y))^{2}\right)}

7611-496: The 1954 Oppenheimer security hearing ("Where there's smoke there's fire") offended him. They separated on May 20, 1956. An interlocutory decree of divorce was entered on June 19, 1956, on the grounds of "extreme cruelty". The divorce became final on May 5, 1958. ... the appointee's wife was granted a divorce from him because of appointee's constantly working calculus problems in his head as soon as awake, while driving car, sitting in living room, and so forth, and that his one hobby

7740-570: The California Institute of Technology (Caltech). Part of the deal was that he could spend his first year on sabbatical in Brazil. He had become smitten by Mary Louise Bell from Neodesha, Kansas . They had met in a cafeteria in Cornell, where she had studied the history of Mexican art and textiles. She later followed him to Caltech, where he gave a lecture. While he was in Brazil, she taught classes on

7869-662: The Feynman–Kac formula is used to efficiently calculate solutions to the Black–Scholes equation to price options on stocks and zero-coupon bond prices in affine term structure models . For example, consider a stock price S t {\displaystyle S_{t}} undergoing geometric Brownian motion d S t = ( r t d t + σ t d W t ) S t {\displaystyle dS_{t}=(r_{t}dt+\sigma _{t}dW_{t})S_{t}} where r t {\displaystyle r_{t}}

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7998-1714: The Feynman–Kac formula, we obtain the Black–Scholes equation: { ∂ t u + A u − r t u = 0 u ( x , T ) = ( x − K ) + {\displaystyle {\begin{cases}\partial _{t}u+Au-r_{t}u=0\\u(x,T)=(x-K)^{+}\end{cases}}} where A = ( r t − σ t 2 / 2 ) ∂ ln ⁡ x + 1 2 σ t 2 ∂ ln ⁡ x 2 = r t x ∂ x + 1 2 σ t 2 x 2 ∂ x 2 . {\textstyle A=(r_{t}-\sigma _{t}^{2}/2)\partial _{\ln x}+{\frac {1}{2}}\sigma _{t}^{2}\partial _{\ln x}^{2}=r_{t}x\partial _{x}+{\frac {1}{2}}\sigma _{t}^{2}x^{2}\partial _{x}^{2}.} More generally, consider an option expiring at time T {\displaystyle T} with payoff g ( S T ) {\displaystyle g(S_{T})} . The same calculation shows that its price u ( x , t ) {\displaystyle u(x,t)} satisfies { ∂ t u + A u − r t u = 0 u ( x , T ) = g ( x ) . {\displaystyle {\begin{cases}\partial _{t}u+Au-r_{t}u=0\\u(x,T)=g(x).\end{cases}}} Some other options like

8127-511: The Talmud had evolved, and everything that was discussed was carefully recorded. Despite being impressed, Feynman was disappointed with the lack of interest for nature and the outside world expressed by the rabbis, who cared about only those questions which arise from the Talmud. Feynman attended Far Rockaway High School , which was also attended by fellow Nobel laureates Burton Richter and Baruch Samuel Blumberg . Upon starting high school, Feynman

8256-630: The University of California, but the head of the physics department, Raymond T. Birge , was reluctant. He made Feynman an offer in May 1945, but Feynman turned it down. Cornell matched its salary offer of $ 3,900 (equivalent to $ 66,000 in 2023) per annum. Feynman became one of the first of the Los Alamos Laboratory's group leaders to depart, leaving for Ithaca, New York , in October 1945. Because Feynman

8385-1354: The above partial differential equation and is therefore zero. What remains is: d Y s = exp ⁡ ( − ∫ t s V ( X τ , τ ) d τ ) σ ( X , s ) ∂ u ∂ X d W . {\displaystyle dY_{s}=\exp \left(-\int _{t}^{s}V(X_{\tau },\tau )\,d\tau \right)\sigma (X,s){\frac {\partial u}{\partial X}}\,dW.} Integrating this equation from t {\displaystyle t} to T {\displaystyle T} , one concludes that: Y ( T ) − Y ( t ) = ∫ t T exp ⁡ ( − ∫ t s V ( X τ , τ ) d τ ) σ ( X , s ) ∂ u ∂ X d W . {\displaystyle Y(T)-Y(t)=\int _{t}^{T}\exp \left(-\int _{t}^{s}V(X_{\tau },\tau )\,d\tau \right)\sigma (X,s){\frac {\partial u}{\partial X}}\,dW.} Upon taking expectations, conditioned on X t = x {\displaystyle X_{t}=x} , and observing that

8514-578: The base of natural logarithms , e = 2.71828 ...), and found that the three filing cabinets where a colleague kept research notes all had the same combination. He left notes in the cabinets as a prank, spooking his colleague, Frederic de Hoffmann , into thinking a spy had gained access to them. Feynman's $ 380 (equivalent to $ 7,000 in 2023) monthly salary was about half the amount needed for his modest living expenses and Arline's medical bills, and they were forced to dip into her $ 3,300 (equivalent to $ 58,000 in 2023) in savings. On weekends he borrowed

8643-560: The beach at Lake Geneva , he met Gweneth Howarth, who was from Ripponden , West Yorkshire, and working in Switzerland as an au pair . Feynman's love life had been turbulent since his divorce; his previous girlfriend had walked off with his Albert Einstein Award medal and, on the advice of an earlier girlfriend, had feigned pregnancy and extorted him into paying for an abortion, then used the money to buy furniture. When Feynman found that Howarth

8772-693: The best-known scientists in the world. In a 1999 poll of 130 leading physicists worldwide by the British journal Physics World , he was ranked the seventh-greatest physicist of all time. He assisted in the development of the atomic bomb during World War II and became known to the wider public in the 1980s as a member of the Rogers Commission , the panel that investigated the Space Shuttle Challenger disaster . Along with his work in theoretical physics, Feynman has been credited with having pioneered

8901-527: The biography Genius: The Life and Science of Richard Feynman by James Gleick . Feynman was born on May 11, 1918, in New York City, to Lucille ( née  Phillips ; 1895–1981), a homemaker, and Melville Arthur Feynman (1890–1946), a sales manager. Feynman's father was born into a Jewish family in Minsk , Russian Empire , and immigrated with his parents to the United States at the age of five. Feynman's mother

9030-489: The cafeteria at Cornell when someone tossed a dinner plate in the air. He read the work of Sir William Rowan Hamilton on quaternions , and tried unsuccessfully to use them to formulate a relativistic theory of electrons. His work during this period, which used equations of rotation to express various spinning speeds, ultimately proved important to his Nobel Prize–winning work, yet because he felt burned out and had turned his attention to less immediately practical problems, he

9159-603: The calutron, but Feynman and Paul Olum struggled to determine whether it was practical. Ultimately, on Lawrence's recommendation, the isotron project was abandoned. At this juncture, in early 1943, Robert Oppenheimer was establishing the Los Alamos Laboratory , a secret laboratory on a mesa in New Mexico where atomic bombs would be designed and built. An offer was made to the Princeton team to be redeployed there. "Like

9288-503: The case that X and Y are not necessarily in L 2 {\displaystyle L^{2}} . The conditional expectation is often approximated in applied mathematics and statistics due to the difficulties in analytically calculating it, and for interpolation. The Hilbert subspace defined above is replaced with subsets thereof by restricting the functional form of g , rather than allowing any measurable function. Examples of this are decision tree regression when g

9417-426: The conditions of Feynman's scholarship to Princeton was that he could not be married; nevertheless, he continued to see his high school sweetheart, Arline Greenbaum, and was determined to marry her once he had been awarded his PhD despite the knowledge that she was seriously ill with tuberculosis . This was an incurable disease at the time, and she was not expected to live more than two years. On June 29, 1942, they took

9546-414: The context of linear regression , this lack of uniqueness is called multicollinearity . Conditional expectation is unique up to a set of measure zero in R n {\displaystyle \mathbb {R} ^{n}} . The measure used is the pushforward measure induced by Y . In the first example, the pushforward measure is a Dirac distribution at 1. In the second it is concentrated on

9675-478: The context, the conditional expectation can be either a random variable or a function. The random variable is denoted E ( X ∣ Y ) {\displaystyle E(X\mid Y)} analogously to conditional probability . The function form is either denoted E ( X ∣ Y = y ) {\displaystyle E(X\mid Y=y)} or a separate function symbol such as f ( y ) {\displaystyle f(y)}

9804-471: The difficulty of placing them." Morse conceded that Feynman was indeed Jewish, but reassured Smyth that Feynman's "physiognomy and manner, however, show no trace of this characteristic". Attendees at Feynman's first seminar, which was on the classical version of the Wheeler–Feynman absorber theory , included Albert Einstein , Wolfgang Pauli , and John von Neumann . Pauli made the prescient comment that

9933-414: The engineers there in devising safety procedures for material storage so that criticality accidents could be avoided, especially when enriched uranium came into contact with water, which acted as a neutron moderator . He insisted on giving the rank and file a lecture on nuclear physics so that they would realize the dangers. He explained that while any amount of unenriched uranium could be safely stored,

10062-401: The enriched uranium had to be carefully handled. He developed a series of safety recommendations for the various grades of enrichments. He was told that if the people at Oak Ridge gave him any difficulty with his proposals, he was to inform them that Los Alamos "could not be responsible for their safety otherwise". Returning to Los Alamos, Feynman was put in charge of the group responsible for

10191-448: The event Y = y . {\displaystyle Y=y.} The conditional expectation of X {\displaystyle X} given Y = y {\displaystyle Y=y} is When the denominator is zero, the expression is undefined. Conditioning on a continuous random variable is not the same as conditioning on the event { Y = y } {\displaystyle \{Y=y\}} as it

10320-597: The expectation of A conditional on B = 1 (i.e., conditional on the die roll being 2, 3, or 5) is E [ A ∣ B = 1 ] = ( 1 + 0 + 0 ) / 3 = 1 / 3 {\displaystyle E[A\mid B=1]=(1+0+0)/3=1/3} , and the expectation of A conditional on B = 0 (i.e., conditional on the die roll being 1, 4, or 6) is E [ A ∣ B = 0 ] = ( 0 + 1 + 1 ) / 3 = 2 / 3 {\displaystyle E[A\mid B=0]=(0+1+1)/3=2/3} . Likewise,

10449-552: The expectation of B conditional on A = 1 is E [ B ∣ A = 1 ] = ( 1 + 0 + 0 ) / 3 = 1 / 3 {\displaystyle E[B\mid A=1]=(1+0+0)/3=1/3} , and the expectation of B conditional on A = 0 is E [ B ∣ A = 0 ] = ( 0 + 1 + 1 ) / 3 = 2 / 3 {\displaystyle E[B\mid A=0]=(0+1+1)/3=2/3} . Suppose we have daily rainfall data (mm of rain each day) collected by

10578-799: The field of quantum computing and introducing the concept of nanotechnology . He held the Richard C. Tolman professorship in theoretical physics at the California Institute of Technology. Feynman was a keen popularizer of physics through both books and lectures, including a 1959 talk on top-down nanotechnology called There's Plenty of Room at the Bottom and the three-volume publication of his undergraduate lectures, The Feynman Lectures on Physics . Feynman also became known through his autobiographical books Surely You're Joking, Mr. Feynman! and What Do You Care What Other People Think? , and books written about him such as Tuva or Bust! by Ralph Leighton and

10707-612: The function X : Ω → R n {\displaystyle X\colon \Omega \to \mathbb {R} ^{n}} is usually not H {\displaystyle {\mathcal {H}}} -measurable, thus the existence of the integrals of the form ∫ H X d P | H {\textstyle \int _{H}X\,dP|_{\mathcal {H}}} , where H ∈ H {\displaystyle H\in {\mathcal {H}}} and P | H {\displaystyle P|_{\mathcal {H}}}

10836-408: The groundwork for the path integral formulation and Feynman diagrams. With Murray Gell-Mann , Feynman developed a model of weak decay , which showed that the current coupling in the process is a combination of vector and axial currents (an example of weak decay is the decay of a neutron into an electron, a proton, and an antineutrino ). Although E. C. George Sudarshan and Robert Marshak developed

10965-648: The help of the conditional expectation. A conditional expectation of X given H {\displaystyle {\mathcal {H}}} , denoted as E ⁡ ( X ∣ H ) {\displaystyle \operatorname {E} (X\mid {\mathcal {H}})} , is any H {\displaystyle {\mathcal {H}}} - measurable function Ω → R n {\displaystyle \Omega \to \mathbb {R} ^{n}} which satisfies: for each H ∈ H {\displaystyle H\in {\mathcal {H}}} . As noted in

11094-469: The history of furniture and interiors at Michigan State University . He proposed to her by mail from Rio de Janeiro, and they married in Boise, Idaho , on June 28, 1952, shortly after he returned. They frequently quarreled and she was frightened by what she decided was "a violent temper". Their politics were different; although he registered and voted as a Republican , she was more conservative, and her opinion on

11223-607: The house's convenience and said that "it's there that I did the fundamental work" for which he won the Nobel Prize. Feynman spent several weeks in Rio de Janeiro in July 1949. That year, the Soviet Union detonated its first atomic bomb , generating concerns about espionage. Fuchs was arrested as a Soviet spy in 1950 and the FBI questioned Bethe about Feynman's loyalty. Physicist David Bohm

11352-690: The integral is taken over all random walks , then I = ∫ w ( x , t ) g ( x ) d x {\displaystyle I=\int w(x,t)g(x)\,dx} where w ( x , t ) is a solution to the parabolic partial differential equation ∂ w ∂ t = 1 2 ∂ 2 w ∂ x 2 − u V ( x ) w {\displaystyle {\frac {\partial w}{\partial t}}={\frac {1}{2}}{\frac {\partial ^{2}w}{\partial x^{2}}}-uV(x)w} with initial condition w ( x , 0) = f ( x ) . In quantitative finance ,

11481-592: The machine to click in musical rhythms. Other work at Los Alamos included calculating neutron equations for the Los Alamos "Water Boiler", a small nuclear reactor , to measure how close an assembly of fissile material was to criticality. On completing this work, Feynman was sent to the Clinton Engineer Works in Oak Ridge, Tennessee , where the Manhattan Project had its uranium enrichment facilities. He aided

11610-418: The native materials of theoretical science. It was not just a facility at mathematics (though it had become clear ... that the mathematical machinery emerging in the Wheeler–Feynman collaboration was beyond Wheeler's own ability). Feynman seemed to possess a frightening ease with the substance behind the equations, like Einstein at the same age, like the Soviet physicist Lev Landau —but few others. One of

11739-474: The need to remedy them". In 1975, in Long Beach, CA , Feynman bought a Dodge Tradesman Maxivan with a bronze-khaki exterior and yellow-green interior, with custom Feynman diagram exterior murals. After Feynman's death, Gweneth sold the van for $ 1 to one of Feynman's friends, film producer Ralph Leighton, who later put it into storage, where it began to rust. In 2012, video game designer Seamus Blackley ,

11868-566: The number of Jews admitted . Instead, he attended the Massachusetts Institute of Technology , where he joined the Pi Lambda Phi fraternity. Although he originally majored in mathematics, he later switched to electrical engineering, as he considered mathematics to be too abstract. Noticing that he "had gone too far", he then switched to physics, which he claimed was "somewhere in between". As an undergraduate, he published two papers in

11997-525: The only university that ever had the good sense to fire me." As early as October 30, 1943, Bethe had written to the chairman of the physics department of his university, Cornell , to recommend that Feynman be hired. On February 28, 1944, this was endorsed by Robert Bacher , also from Cornell, and one of the most senior scientists at Los Alamos. This led to an offer being made in August 1944, which Feynman accepted. Oppenheimer had also hoped to recruit Feynman to

12126-399: The particle incur "cost" at a rate of f ( X s , s ) {\displaystyle f(X_{s},s)} at location X s {\displaystyle X_{s}} at time s {\displaystyle s} . Let it incur a final cost at ψ ( X T ) {\displaystyle \psi (X_{T})} . Also, allow

12255-422: The particle to decay. If the particle is at location X s {\displaystyle X_{s}} at time s {\displaystyle s} , then it decays with rate V ( X s , s ) {\displaystyle V(X_{s},s)} . After the particle has decayed, all future cost is zero. Then u ( x , t ) {\displaystyle u(x,t)}

12384-459: The powerful new tool that Feynman had created. Computer programs were later written to evaluate Feynman diagrams, enabling physicists to use quantum field theory to make high-precision predictions . Marc Kac adapted Feynman's technique of summing over possible histories of a particle to the study of parabolic partial differential equations , yielding what is now known as the Feynman–Kac formula ,

12513-524: The project and was present at the Trinity nuclear test . Feynman claimed to be the only person to see the explosion without the very dark glasses or welder's lenses provided, reasoning that it was safe to look through a truck windshield, as it would screen out the harmful ultraviolet radiation. The immense brightness of the explosion made him duck to the truck's floor, where he saw a temporary "purple splotch" afterimage . Feynman nominally held an appointment at

12642-499: The question showed that the superfluid was displaying quantum mechanical behavior observable on a macroscopic scale. This helped with the problem of superconductivity , but the solution eluded Feynman. It was solved with the BCS theory of superconductivity, proposed by John Bardeen , Leon Neil Cooper , and John Robert Schrieffer in 1957. Feynman, inspired by a desire to quantize the Wheeler–Feynman absorber theory of electrodynamics, laid

12771-1343: The real-valued case of Feynman's path integrals . The complex case, which occurs when a particle's spin is included, is still an open question. It offers a method of solving certain partial differential equations by simulating random paths of a stochastic process. Conversely, an important class of expectations of random processes can be computed by deterministic methods. Consider the partial differential equation ∂ ∂ t u ( x , t ) + μ ( x , t ) ∂ ∂ x u ( x , t ) + 1 2 σ 2 ( x , t ) ∂ 2 ∂ x 2 u ( x , t ) − V ( x , t ) u ( x , t ) + f ( x , t ) = 0 , {\displaystyle {\frac {\partial }{\partial t}}u(x,t)+\mu (x,t){\frac {\partial }{\partial x}}u(x,t)+{\tfrac {1}{2}}\sigma ^{2}(x,t){\frac {\partial ^{2}}{\partial x^{2}}}u(x,t)-V(x,t)u(x,t)+f(x,t)=0,} defined for all x ∈ R {\displaystyle x\in \mathbb {R} } and t ∈ [ 0 , T ] {\displaystyle t\in [0,T]} , subject to

12900-2150: The right side is an Itô integral , which has expectation zero, it follows that: E [ Y ( T ) ∣ X t = x ] = E [ Y ( t ) ∣ X t = x ] = u ( x , t ) . {\displaystyle E[Y(T)\mid X_{t}=x]=E[Y(t)\mid X_{t}=x]=u(x,t).} The desired result is obtained by observing that: E [ Y ( T ) ∣ X t = x ] = E [ exp ⁡ ( − ∫ t T V ( X τ , τ ) d τ ) u ( X T , T ) + ∫ t T exp ⁡ ( − ∫ t r V ( X τ , τ ) d τ ) f ( X r , r ) d r | X t = x ] {\displaystyle E[Y(T)\mid X_{t}=x]=E\left[\exp \left(-\int _{t}^{T}V(X_{\tau },\tau )\,d\tau \right)u(X_{T},T)+\int _{t}^{T}\exp \left(-\int _{t}^{r}V(X_{\tau },\tau )\,d\tau \right)f(X_{r},r)\,dr\,{\Bigg |}\,X_{t}=x\right]} and finally u ( x , t ) = E [ exp ⁡ ( − ∫ t T V ( X τ , τ ) d τ ) ψ ( X T ) + ∫ t T exp ⁡ ( − ∫ t s V ( X τ , τ ) d τ ) f ( X s , s ) d s | X t = x ] {\displaystyle u(x,t)=E\left[\exp \left(-\int _{t}^{T}V(X_{\tau },\tau )\,d\tau \right)\psi (X_{T})+\int _{t}^{T}\exp \left(-\int _{t}^{s}V(X_{\tau },\tau )\,d\tau \right)f(X_{s},s)\,ds\,{\Bigg |}\,X_{t}=x\right]} The Feynman–Kac formula can also be interpreted as

13029-604: The summer working on ballistics problems at the Frankford Arsenal in Pennsylvania . After the attack on Pearl Harbor brought the United States into the war, Feynman was recruited by Robert R. Wilson , who was working on means to produce enriched uranium for use in an atomic bomb , as part of what would become the Manhattan Project . At the time, Feynman had not earned a graduate degree. Wilson's team at Princeton

13158-455: The ten–year period that fall in March. Similarly, the conditional expectation of rainfall conditional on days dated March 2 is the average of the rainfall amounts that occurred on the ten days with that specific date. The related concept of conditional probability dates back at least to Laplace , who calculated conditional distributions. It was Andrey Kolmogorov who, in 1933, formalized it using

13287-522: The terminal condition u ( x , T ) = ψ ( x ) , {\displaystyle u(x,T)=\psi (x),} where μ , σ , ψ , V , f {\displaystyle \mu ,\sigma ,\psi ,V,f} are known functions, T {\displaystyle T} is a parameter, and u : R × [ 0 , T ] → R {\displaystyle u:\mathbb {R} \times [0,T]\to \mathbb {R} }

13416-706: The theoretical work and calculations on the proposed uranium hydride bomb , which ultimately proved to be infeasible. He was sought out by physicist Niels Bohr for one-on-one discussions. He later discovered the reason: most of the other physicists were too much in awe of Bohr to argue with him. Feynman had no such inhibitions, vigorously pointing out anything he considered to be flawed in Bohr's thinking. He said he felt as much respect for Bohr as anyone else, but once anyone got him talking about physics, he would become so focused he forgot about social niceties. Perhaps because of this, Bohr never warmed to Feynman. At Los Alamos, which

13545-504: The theoreticians were completely overshadowed by the achievements of the experimentalists, who reported the discovery of the Lamb shift , the measurement of the magnetic moment of the electron, and Robert Marshak 's two-meson hypothesis. Bethe took the lead from the work of Hans Kramers , and derived a renormalized non-relativistic quantum equation for the Lamb shift. The next step was to create

13674-436: The theory nearly simultaneously, Feynman's collaboration with Gell-Mann was seen as seminal because the weak interaction was neatly described by the vector and axial currents. It thus combined the 1933 beta decay theory of Enrico Fermi with an explanation of parity violation . Feynman attempted an explanation, called the parton model , of the strong interactions governing nucleon scattering. The parton model emerged as

13803-568: The theory would be extremely difficult to quantize, and Einstein said that one might try to apply this method to gravity in general relativity , which Sir Fred Hoyle and Jayant Narlikar did much later as the Hoyle–Narlikar theory of gravity . Feynman received a PhD from Princeton in 1942; his thesis advisor was John Archibald Wheeler . In his doctoral thesis entitled "The Principle of Least Action in Quantum Mechanics", Feynman applied

13932-482: The theory, which Feynman and Wheeler had tried, unsuccessfully, to work around. "Theoreticians", noted Murray Gell-Mann , "were in disgrace". In June 1947, leading American physicists met at the Shelter Island Conference . For Feynman, it was his "first big conference with big men ... I had never gone to one like this one in peacetime." The problems plaguing quantum electrodynamics were discussed, but

14061-2532: The third term is O ( d t d u ) {\displaystyle O(dt\,du)} and can be dropped. We also have that d ( ∫ t s exp ⁡ ( − ∫ t r V ( X τ , τ ) d τ ) f ( X r , r ) d r ) = exp ⁡ ( − ∫ t s V ( X τ , τ ) d τ ) f ( X s , s ) d s . {\displaystyle d\left(\int _{t}^{s}\exp \left(-\int _{t}^{r}V(X_{\tau },\tau )\,d\tau \right)f(X_{r},r)dr\right)=\exp \left(-\int _{t}^{s}V(X_{\tau },\tau )\,d\tau \right)f(X_{s},s)ds.} Applying Itô's lemma to d u ( X s , s ) {\displaystyle du(X_{s},s)} , it follows that d Y s = exp ⁡ ( − ∫ t s V ( X τ , τ ) d τ ) ( − V ( X s , s ) u ( X s , s ) + f ( X s , s ) + μ ( X s , s ) ∂ u ∂ X + ∂ u ∂ s + 1 2 σ 2 ( X s , s ) ∂ 2 u ∂ X 2 ) d s + exp ⁡ ( − ∫ t s V ( X τ , τ ) d τ ) σ ( X , s ) ∂ u ∂ X d W . {\displaystyle {\begin{aligned}dY_{s}={}&\exp \left(-\int _{t}^{s}V(X_{\tau },\tau )\,d\tau \right)\,\left(-V(X_{s},s)u(X_{s},s)+f(X_{s},s)+\mu (X_{s},s){\frac {\partial u}{\partial X}}+{\frac {\partial u}{\partial s}}+{\tfrac {1}{2}}\sigma ^{2}(X_{s},s){\frac {\partial ^{2}u}{\partial X^{2}}}\right)\,ds\\[6pt]&{}+\exp \left(-\int _{t}^{s}V(X_{\tau },\tau )\,d\tau \right)\sigma (X,s){\frac {\partial u}{\partial X}}\,dW.\end{aligned}}} The first term contains, in parentheses,

14190-509: The use of which extends beyond physics to many applications of stochastic processes . To Schwinger, however, the Feynman diagram was "pedagogy, not physics". By 1949, Feynman was becoming restless at Cornell. He never settled into a particular house or apartment, living in guest houses or student residences, or with married friends "until these arrangements became sexually volatile". He liked to date undergraduates, hire prostitutes, and sleep with

14319-665: The wives of friends. He was not fond of Ithaca's cold winter weather, and pined for a warmer climate. Above all, he was always in the shadow of Hans Bethe at Cornell. Despite all of this, Feynman looked back favorably on the Telluride House , where he resided for a large period of his Cornell career. In an interview, he described the House as "a group of boys that have been specially selected because of their scholarship, because of their cleverness or whatever it is, to be given free board and lodging and so on, because of their brains". He enjoyed

14448-482: Was arrested on December 4, 1950 and emigrated to Brazil in October 1951. Because of the fears of a nuclear war, a girlfriend told Feynman that he should also consider moving to South America. He had a sabbatical coming for 1951–1952, and elected to spend it in Brazil, where he gave courses at the Centro Brasileiro de Pesquisas Físicas . In Brazil, Feynman was impressed with samba music, and learned to play

14577-553: Was being paid only $ 25 a month, he offered her $ 20 (equivalent to $ 202 in 2022) a week to be his live-in maid. Feynman knew that this sort of behavior was illegal under the Mann Act , so he had a friend, Matthew Sands , act as her sponsor. Howarth pointed out that she already had two boyfriends, but decided to take Feynman up on his offer, and arrived in Altadena, California , in June 1959. She made

14706-422: Was born and the family moved to Far Rockaway, Queens . Though separated by nine years, Joan and Richard were close, and they both shared a curiosity about the world. Though their mother thought women lacked the capacity to understand such things, Richard encouraged Joan's interest in astronomy, and Joan eventually became an astrophysicist. Feynman's parents were both from Jewish families, and his family went to

14835-554: Was born in the United States into a Jewish family. Lucille's father had emigrated from Poland, and her mother also came from a family of Polish immigrants. She trained as a primary school teacher but married Melville in 1917, before taking up a profession. Feynman was a late talker and did not speak until after his third birthday. As an adult, he spoke with a New York accent strong enough to be perceived as an affectation or exaggeration, so much so that his friends Wolfgang Pauli and Hans Bethe once commented that Feynman spoke like

14964-416: Was clear: Shin'ichirō Tomonaga , Schwinger and Feynman understood what they were talking about even if no one else did, but had not published anything. He was convinced that Feynman's formulation was easier to understand, and ultimately managed to convince Oppenheimer that this was the case. Dyson published a paper in 1949, which added new rules to Feynman's that told how to implement renormalization. Feynman

15093-451: Was experimenting with mathematical topics such as the half-derivative using his own notation. He created special symbols for logarithm , sine , cosine and tangent functions so they did not look like three variables multiplied together, and for the derivative , to remove the temptation of canceling out the d {\displaystyle d} 's in d / d x {\displaystyle d/dx} . A member of

15222-474: Was in the discrete case. For a discussion, see Conditioning on an event of probability zero . Not respecting this distinction can lead to contradictory conclusions as illustrated by the Borel-Kolmogorov paradox . All random variables in this section are assumed to be in L 2 {\displaystyle L^{2}} , that is square integrable . In its full generality, conditional expectation

15351-404: Was isolated for security, Feynman amused himself by investigating the combination locks on the cabinets and desks of physicists. He often found that they left the lock combinations on the factory settings, wrote the combinations down, or used easily guessable combinations like dates. He found one cabinet's combination by trying numbers he thought a physicist might use (it proved to be 27–18–28 after

15480-486: Was no longer working at the Los Alamos Laboratory, he was no longer exempt from the draft . At his induction physical, Army psychiatrists diagnosed Feynman as suffering from a mental illness and the Army gave him a 4-F exemption on mental grounds. His father died suddenly on October 8, 1946, and Feynman suffered from depression. On October 17, 1946, he wrote a letter to Arline, expressing his deep love and heartbreak. The letter

15609-465: Was not central to the project. He administered the computation group of human computers in the theoretical division. With Stanley Frankel and Nicholas Metropolis , he assisted in establishing a system for using IBM punched cards for computation. He invented a new method of computing logarithms that he later used on the Connection Machine . An avid drummer, Feynman figured out how to get

15738-593: Was not only converted to other religious views, but I also stopped believing that the Jewish people are in any way 'the chosen people '". Later in life, during a visit to the Jewish Theological Seminary , Feynman encountered the Talmud for the first time. He saw that it contained the original text in a little square on each page, and surrounding it were commentaries written over time by different people. In this way

15867-410: Was playing his African drums. His ex-wife reportedly testified that on several occasions when she unwittingly disturbed either his calculus or his drums he flew into a violent rage, during which time he choked her, threw pieces of bric-a-brac about and smashed the furniture ... In the wake of the 1957 Sputnik crisis , the U.S. government's interest in science rose for a time. Feynman was considered for

15996-499: Was probably the first job Feynman had, and during this time he showed early signs of an aptitude for his later career in theoretical physics, when he would analyze the issues theoretically and arrive at the solutions. When he was in grade school, he created a home burglar alarm system while his parents were out for the day running errands. When Richard was five, his mother gave birth to a younger brother, Henry Phillips, who died at age four weeks. Four years later, Richard's sister Joan

16125-536: Was prompted to publish his ideas in the Physical Review in a series of papers over three years. His 1948 papers on "A Relativistic Cut-Off for Classical Electrodynamics" attempted to explain what he had been unable to get across at Pocono. His 1949 paper on "The Theory of Positrons" addressed the Schrödinger equation and Dirac equation , and introduced what is now called the Feynman propagator . Finally, in papers on

16254-635: Was quickly promoted to a higher math class. An IQ test administered in high school estimated his IQ at 125—high but "merely respectable", according to biographer James Gleick . His sister Joan, who scored one point higher, later jokingly claimed to an interviewer that she was smarter. Years later he declined to join Mensa International , saying that his IQ was too low. When Feynman was 15, he taught himself trigonometry , advanced algebra , infinite series , analytic geometry , and both differential and integral calculus . Before entering college, he

16383-399: Was sealed and only opened after his death. "Please excuse my not mailing this," the letter concluded, "but I don't know your new address." Unable to focus on research problems, Feynman began tackling physics problems, not for utility, but for self-satisfaction. One of these involved analyzing the physics of a twirling, nutating disk as it is moving through the air, inspired by an incident in

16512-483: Was surprised by the offers of professorships from other renowned universities, including the Institute for Advanced Study , the University of California, Los Angeles , and the University of California, Berkeley . Feynman was not the only frustrated theoretical physicist in the early post-war years. Quantum electrodynamics suffered from infinite integrals in perturbation theory . These were clear mathematical flaws in

16641-465: Was working on a device called an isotron, intended to electromagnetically separate uranium-235 from uranium-238 . This was done in a quite different manner from that used by the calutron that was under development by a team under Wilson's former mentor, Ernest O. Lawrence , at the Radiation Laboratory of the University of California . On paper, the isotron was many times more efficient than

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