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56-649: In hole theory, the solutions with negative time evolution factors are reinterpreted as representing the positron, discovered by Carl Anderson . The interpretation of this result requires a Dirac sea, showing that the Dirac equation is not merely a combination of special relativity and quantum mechanics , but it also implies that the number of particles cannot be conserved. Dirac sea theory has been displaced by quantum field theory , though they are mathematically compatible. Similar ideas on holes in crystals had been developed by Soviet physicist Yakov Frenkel in 1926, but there

112-435: A † ( k ) {\displaystyle a^{\dagger }(k)} gives zero when the state with momentum k is already filled, while the annihilation operator a ( k ) {\displaystyle a(k)} gives zero when the state with momentum k is empty. But then it is possible to reinterpret the annihilation operator as a creation operator for a negative energy particle. It still lowers

168-422: A binary number in which ones are used as separators at the start of each layer, while a number within a given layer (such as a guest's coach number) is represented with that many zeroes. Thus, a guest with the prior address 2-5-1-3-1 (five infinite layers) would go to room 10010000010100010 (decimal 295458). As an added step in this process, one zero can be removed from each section of the number; in this example,

224-404: A chiral anomaly and a gauge instanton . The development of quantum field theory (QFT) in the 1930s made it possible to reformulate the Dirac equation in a way that treats the positron as a "real" particle rather than the absence of a particle, and makes the vacuum the state in which no particles exist instead of an infinite sea of particles. This picture recaptures all the valid predictions of

280-435: A muon (or 'mu-meson', as it was known for many years), a subatomic particle 207 times more massive than the electron , but with the same negative electric charge and spin 1/2 as the electron, again in cosmic rays . Anderson and Neddermeyer at first believed that they had seen a pion , a particle which Hideki Yukawa had postulated in his theory of the strong interaction . When it became clear that what Anderson had seen

336-451: A particle with the same mass as the electron , but with opposite electrical charge . This discovery, announced in 1932 and later confirmed by others, validated Paul Dirac 's theoretical prediction of the existence of the positron . Anderson first detected the particles in cosmic rays . He then produced more conclusive proof by shooting gamma rays produced by the natural radioactive nuclide ThC'' ( Tl ) into other materials, resulting in

392-413: A preexisting space-time, making it possible to realize the concept that space-time and all structures therein arise as a result of the collective interaction of the sea states with each other and with the additional particles and "holes" in the sea. Carl David Anderson Carl David Anderson (September 3, 1905 – January 11, 1991) was an American physicist . He is best known for his discovery of

448-467: A process that could continue without limit as the electron descends into ever lower energy states. However, real electrons clearly do not behave in this way. Dirac's solution to this was to rely on the Pauli exclusion principle . Electrons are fermions , and obey the exclusion principle, which means that no two electrons can share a single energy state within an atom. Dirac hypothesized that what we think of as

504-473: A rather peculiar feature: for each quantum state possessing a positive energy E , there is a corresponding state with energy - E . This is not a big difficulty when an isolated electron is considered, because its energy is conserved and negative-energy electrons may be left out. However, difficulties arise when effects of the electromagnetic field are considered, because a positive-energy electron would be able to shed energy by continuously emitting photons ,

560-410: A room to go to. After this, room 1 is empty and the new guest can be moved into that room. By repeating this procedure, it is possible to make room for any finite number of new guests. In general, when k guests seek a room, the hotel can apply the same procedure and move every guest from room n to room n + k . It is also possible to accommodate a countably infinite number of new guests: just move

616-411: A situation might exist in which all the negative-energy states are occupied except one. This "hole" in the sea of negative-energy electrons would respond to electric fields as though it were a positively charged particle. Initially, Dirac identified this hole as a proton . However, Robert Oppenheimer pointed out that an electron and its hole would be able to annihilate each other, releasing energy on

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672-647: A unique prime factorization , it is easy to see all people will have a room, while no two people will end up in the same room. For example, the person in room 2592 ( 2 5 3 4 {\displaystyle 2^{5}3^{4}} ) was sitting in on the 4th coach, on the 5th seat. Like the prime powers method, this solution leaves certain rooms empty. This method can also easily be expanded for infinite nights, infinite entrances, etc. ( 2 s 3 c 5 n 7 e . . . {\displaystyle 2^{s}3^{c}5^{n}7^{e}...} ) For each passenger, compare

728-422: Is a constant shift in quantities like the energy and the charge density, quantities that count the total number of particles. The infinite constant gives the Dirac sea an infinite energy and charge density. The vacuum charge density should be zero, since the vacuum is Lorentz invariant , but this is artificial to arrange in Dirac's picture. The way it is done is by passing to the modern interpretation. Dirac's idea

784-400: Is a direct consequence of the relativistic energy-momentum relation E 2 = p 2 c 2 + m 2 c 4 {\displaystyle E^{2}=p^{2}c^{2}+m^{2}c^{4}} upon which the Dirac equation is built. The quantity U is a constant 2 × 1 column vector and N is a normalization constant. The quantity ε is called

840-396: Is a situation involving three "levels" of infinity , and it can be solved by extensions of any of the previous solutions. The prime factorization method can be applied by adding a new prime number for every additional layer of infinity ( 2 s 3 c 5 f {\displaystyle 2^{s}3^{c}5^{f}} , with f {\displaystyle f}

896-588: Is arbitrary, and the roles of the two numbers can be reversed (seat-odd and coach-even), so long as it is applied consistently. Those already in the hotel will be moved to room ( n 2 + n ) / 2 {\displaystyle (n^{2}+n)/2} , or the n {\displaystyle n} th triangular number . Those in a coach will be in room ( ( c + n − 1 ) 2 + c + n − 1 ) / 2 + n {\displaystyle ((c+n-1)^{2}+c+n-1)/2+n} , or

952-403: Is countable, hence we may enumerate its elements s 1 , s 2 , … {\displaystyle s_{1},s_{2},\dots } . Now if s n = ( a , b ) {\displaystyle s_{n}=(a,b)} , assign the b {\displaystyle b} th guest of the a {\displaystyle a} th coach to

1008-521: Is full. However, it can be shown that the existing guests and newcomers — even an infinite number of them — can each have their own room in the infinite hotel. With one additional guest, the hotel can accommodate him or her and the existing guests if infinitely many guests simultaneously move rooms. The guest currently in room 1 moves to room 2, the guest currently in room 2 to room 3, and so on, moving every guest from their current room n to room n +1. The infinite hotel has no final room, so every guest has

1064-400: Is made, it must be applied uniformly throughout.) Each person of a certain seat s {\displaystyle s} and coach c {\displaystyle c} can be put into room 2 s 3 c {\displaystyle 2^{s}3^{c}} (presuming c =0 for the people already in the hotel, 1 for the first coach, etc.). Because every number has

1120-411: Is more directly applicable to solid state physics , where the valence band in a solid can be regarded as a "sea" of electrons. Holes in this sea indeed occur, and are extremely important for understanding the effects of semiconductors , though they are never referred to as "positrons". Unlike in particle physics, there is an underlying positive charge—the charge of the ionic lattice —that cancels out

1176-530: Is no indication the concept was discussed with Dirac when the two met in a Soviet physics congress in the summer of 1928. The origins of the Dirac sea lie in the energy spectrum of the Dirac equation , an extension of the Schrödinger equation consistent with special relativity , an equation that Dirac had formulated in 1928. Although this equation was extremely successful in describing electron dynamics, it possesses

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1232-430: Is rooms 2 and 3; and so on. The column formed by the set of rightmost rooms will correspond to the triangular numbers. Once they are filled (by the hotel's redistributed occupants), the remaining empty rooms form the shape of a pyramid exactly identical to the original shape. Thus, the process can be repeated for each infinite set. Doing this one at a time for each coach would require an infinite number of steps, but by using

1288-451: Is that there is a nonuniformity in certain expressions, because replacing annihilation with creation adds a constant to the negative energy particle number. The number operator for a Fermi field is: N = a † a = 1 − a a † {\displaystyle N=a^{\dagger }a=1-aa^{\dagger }} which means that if one replaces N by 1− N for negative energy states, there

1344-434: Is unobservable—the cosmological constant aside—the infinite energy density of the vacuum does not represent a problem. Only changes in the energy density are observable. Geoffrey Landis also notes that Pauli exclusion does not definitively mean that a filled Dirac sea cannot accept more electrons, since, as Hilbert elucidated, a sea of infinite extent can accept new particles even if it is filled. This happens when we have

1400-425: The ( c + n − 1 ) {\displaystyle (c+n-1)} triangular number plus n {\displaystyle n} . In this way all the rooms will be filled by one, and only one, guest. This pairing function can be demonstrated visually by structuring the hotel as a one-room-deep, infinitely tall pyramid . The pyramid's topmost row is a single room: room 1; its second row

1456-469: The n {\displaystyle n} th room (consider the guests already in the hotel as guests of the 0 {\displaystyle 0} th coach). Thus we have a function assigning each person to a room; furthermore, this assignment does not skip over any rooms. Suppose the hotel is next to an ocean, and an infinite number of car ferries arrive, each bearing an infinite number of coaches, each with an infinite number of passengers. This

1512-691: The positron in 1932, an achievement for which he received the 1936 Nobel Prize in Physics , and of the muon in 1936. Anderson was born in New York City , the son of Swedish immigrants. He studied physics and engineering at Caltech ( B.S. , 1927; Ph.D. , 1930). Under the supervision of Robert A. Millikan , he began investigations into cosmic rays during the course of which he encountered unexpected particle tracks in his (modern versions now commonly referred to as an Anderson) cloud chamber photographs that he correctly interpreted as having been created by

1568-460: The time evolution factor , and its interpretation in similar roles in, for example, the plane wave solutions of the Schrödinger equation , is the energy of the wave (particle). This interpretation is not immediately available here since it may acquire negative values. A similar situation prevails for the Klein–;Gordon equation . In that case, the absolute value of ε can be interpreted as

1624-443: The " vacuum " is actually the state in which all the negative- energy states are filled, and none of the positive-energy states. Therefore, if we want to introduce a single electron, we would have to put it in a positive-energy state, as all the negative-energy states are occupied. Furthermore, even if the electron loses energy by emitting photons it would be forbidden from dropping below zero energy. Dirac further pointed out that

1680-404: The Dirac hole. Despite its success, the idea of the Dirac sea tends not to strike people as very elegant. The existence of the sea implies an infinite negative electric charge filling all of space. In order to make any sense out of this, one must assume that the "bare vacuum" must have an infinite positive charge density which is exactly cancelled by the Dirac sea. Since the absolute energy density

1736-1921: The Dirac sea, such as electron-positron annihilation. On the other hand, the field formulation does not eliminate all the difficulties raised by the Dirac sea; in particular the problem of the vacuum possessing infinite energy . Upon solving the free Dirac equation, i ℏ ∂ Ψ ∂ t = ( c α ^ ⋅ p ^ + m c 2 β ^ ) Ψ , {\displaystyle i\hbar {\frac {\partial \Psi }{\partial t}}=(c{\hat {\boldsymbol {\alpha }}}\cdot {\hat {\boldsymbol {p}}}+mc^{2}{\hat {\beta }})\Psi ,} one finds Ψ p λ = N ( U ( c σ ^ ⋅ p ) m c 2 + λ E p U ) exp ⁡ [ i ( p ⋅ x − ε t ) / ℏ ] 2 π ℏ 3 , {\displaystyle \Psi _{\mathbf {p} \lambda }=N\left({\begin{matrix}U\\{\frac {(c{\hat {\boldsymbol {\sigma }}}\cdot {\boldsymbol {p}})}{mc^{2}+\lambda E_{p}}}U\end{matrix}}\right){\frac {\exp[i(\mathbf {p} \cdot \mathbf {x} -\varepsilon t)/\hbar ]}{{\sqrt {2\pi \hbar }}^{3}}},} where ε = ± E p , E p = + c p 2 + m 2 c 2 , λ = sgn ⁡ ε {\displaystyle \varepsilon =\pm E_{p},\quad E_{p}=+c{\sqrt {\mathbf {p} ^{2}+m^{2}c^{2}}},\quad \lambda =\operatorname {sgn} \varepsilon } for plane wave solutions with 3 -momentum p . This

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1792-534: The Grand Hotel Hilbert's paradox of the Grand Hotel ( colloquial : Infinite Hotel Paradox or Hilbert's Hotel ) is a thought experiment which illustrates a counterintuitive property of infinite sets . It is demonstrated that a fully occupied hotel with infinitely many rooms may still accommodate additional guests, even infinitely many of them, and this process may be repeated infinitely often. The idea

1848-407: The address 2-3-2 would go to room 232, while the one with the address 4935-198-82217 would go to room #008,402,912,391,587 (the leading zeroes can be removed). Anticipating the possibility of any number of layers of infinite guests, the hotel may wish to assign rooms such that no guest will need to move, no matter how many guests arrive afterward. One solution is to convert each arrival's address into

1904-434: The confusing "zoo" fit into some tidy conceptual scheme. Willis Lamb , in his 1955 Nobel Prize Lecture, joked that he had heard it said that "the finder of a new elementary particle used to be rewarded by a Nobel Prize, but such a discovery now ought to be punished by a 10,000 dollar fine." Anderson spent all of his academic and research career at Caltech . During World War II , he conducted research in rocketry there. He

1960-547: The creation and annihilation operators of two different free field theories. In the modern interpretation, the field operator for a Dirac spinor is a sum of creation operators and annihilation operators, in a schematic notation: ψ ( x ) = ∑ a † ( k ) e i k x + a ( k ) e − i k x {\displaystyle \psi (x)=\sum a^{\dagger }(k)e^{ikx}+a(k)e^{-ikx}} An operator with negative frequency lowers

2016-460: The creation of positron-electron pairs. For this work, Anderson shared the 1936 Nobel Prize in Physics with Victor Hess . Fifty years later, Anderson acknowledged that his discovery was inspired by the work of his Caltech classmate Chung-Yao Chao , whose research formed the foundation from which much of Anderson's work developed but was not credited at the time. Also in 1936, Anderson and his first graduate student, Seth Neddermeyer , discovered

2072-440: The electric charge of the sea. Dirac's original concept of a sea of particles was revived in the theory of causal fermion systems , a recent proposal for a unified physical theory. In this approach, the problems of the infinite vacuum energy and infinite charge density of the Dirac sea disappear because these divergences drop out of the physical equations formulated via the causal action principle . These equations do not require

2128-405: The energy of any state by an amount proportional to the frequency, while operators with positive frequency raise the energy of any state. In the modern interpretation, the positive frequency operators add a positive energy particle, adding to the energy, while the negative frequency operators annihilate a positive energy particle, and lower the energy. For a fermionic field , the creation operator

2184-438: The energy of the vacuum, but in this point of view it does so by creating a negative energy object. This reinterpretation only affects the philosophy. To reproduce the rules for when annihilation in the vacuum gives zero, the notion of "empty" and "filled" must be reversed for the negative energy states. Instead of being states with no antiparticle, these are states that are already filled with a negative energy particle. The price

2240-440: The energy of the wave since in the canonical formalism, waves with negative ε actually have positive energy E p . But this is not the case with the Dirac equation. The energy in the canonical formalism associated with negative ε is – E p . The Dirac sea interpretation and the modern QFT interpretation are related by what may be thought of as a very simple Bogoliubov transformation , an identification between

2296-560: The ferry). The prime power solution can be applied with further exponentiation of prime numbers, resulting in very large room numbers even given small inputs. For example, the passenger in the second seat of the third bus on the second ferry (address 2-3-2) would raise the 2nd odd prime (5) to 49, which is the result of the 3rd odd prime (7) being raised to the power of his seat number (2). This room number would have over thirty decimal digits. The interleaving method can be used with three interleaved "strands" instead of two. The passenger with

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2352-829: The guest's new room is 101000011001 (decimal 2585). This ensures that every room could be filled by a hypothetical guest. If no infinite sets of guests arrive, then only rooms that are a power of two will be occupied. Hilbert's paradox is a veridical paradox : it leads to a counter-intuitive result that is provably true. The statements "there is a guest to every room" and "no more guests can be accommodated" are not equivalent when there are infinitely many rooms. Initially, this state of affairs might seem to be counter-intuitive. The properties of infinite collections of things are quite different from those of finite collections of things. The paradox of Hilbert's Grand Hotel can be understood by using Cantor's theory of transfinite numbers . Thus, in an ordinary (finite) hotel with more than one room,

2408-715: The lengths of n {\displaystyle n} and c {\displaystyle c} as written in any positional numeral system , such as decimal . (Treat each hotel resident as being in coach #0.) If either number is shorter, add leading zeroes to it until both values have the same number of digits. Interleave the digits to produce a room number: its digits will be [first digit of coach number]-[first digit of seat number]-[second digit of coach number]-[second digit of seat number]-etc. The hotel (coach #0) guest in room number 1729 moves to room 01070209 (i.e., room 1,070,209). The passenger on seat 1234 of coach 789 goes to room 01728394 (i.e., room 1,728,394). Unlike

2464-468: The number of arrivals is less than or equal to the number of vacancies created. It is easier to show, by an independent means, that the number of arrivals is also greater than or equal to the number of vacancies, and thus that they are equal , than to modify the algorithm to an exact fit.) (The algorithm works equally well if one interchanges n {\displaystyle n} and c {\displaystyle c} , but whichever choice

2520-479: The number of odd-numbered rooms is obviously smaller than the total number of rooms. However, in Hilbert's Grand Hotel, the quantity of odd-numbered rooms is not smaller than the total "number" of rooms. In mathematical terms, the cardinality of the subset containing the odd-numbered rooms is the same as the cardinality of the set of all rooms. Indeed, infinite sets are characterized as sets that have proper subsets of

2576-424: The order of the electron's rest energy in the form of energetic photons; if holes were protons, stable atoms would not exist. Hermann Weyl also noted that a hole should act as though it has the same mass as an electron, whereas the proton is about two thousand times heavier. The issue was finally resolved in 1932, when the positron was discovered by Carl Anderson , with all the physical properties predicted for

2632-409: The person occupying room 1 to room 2, the guest occupying room 2 to room 4, and, in general, the guest occupying room n to room 2 n (2 times n ), and all the odd-numbered rooms (which are countably infinite) will be free for the new guests. It is possible to accommodate countably infinitely many coachloads of countably infinite passengers each, by several different methods. Most methods depend on

2688-406: The prime powers solution, this one fills the hotel completely, and we can reconstruct a guest's original coach and seat by reversing the interleaving process. First add a leading zero if the room has an odd number of digits. Then de-interleave the number into two numbers: the coach number consists of the odd-numbered digits and the seat number is the even-numbered ones. Of course, the original encoding

2744-451: The prior formulas, a guest can determine what their room "will be" once their coach has been reached in the process, and can simply go there immediately. Let S := { ( a , b ) ∣ a , b ∈ N } {\displaystyle S:=\{(a,b)\mid a,b\in \mathbb {N} \}} . S {\displaystyle S} is countable since N {\displaystyle \mathbb {N} }

2800-459: The rooms p c n {\displaystyle p_{c}^{n}} where p c {\displaystyle p_{c}} is the c {\displaystyle c} th odd prime number . This solution leaves certain rooms empty (which may or may not be useful to the hotel); specifically, all numbers that are not prime powers , such as 15 or 847, will no longer be occupied. (So, strictly speaking, this shows that

2856-427: The same cardinality. For countable sets (sets with the same cardinality as the natural numbers ) this cardinality is ℵ 0 {\displaystyle \aleph _{0}} . Rephrased, for any countably infinite set, there exists a bijective function which maps the countably infinite set to the set of natural numbers, even if the countably infinite set contains the natural numbers. For example,

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2912-490: The seats in the coaches being already numbered (or use the axiom of countable choice ). In general any pairing function can be used to solve this problem. For each of these methods, consider a passenger's seat number on a coach to be n {\displaystyle n} , and their coach number to be c {\displaystyle c} , and the numbers n {\displaystyle n} and c {\displaystyle c} are then fed into

2968-470: The two arguments of the pairing function . Send the guest in room i {\displaystyle i} to room 2 i {\displaystyle 2^{i}} , then put the first coach's load in rooms 3 n {\displaystyle 3^{n}} , the second coach's load in rooms 5 n {\displaystyle 5^{n}} ; in general for coach number c {\displaystyle c} we use

3024-420: Was not the pion, the physicist I. I. Rabi , puzzled as to how the unexpected discovery could fit into any logical scheme of particle physics , quizzically asked "Who ordered that ?" (sometimes the story goes that he was dining with colleagues at a Chinese restaurant at the time). The muon was the first of a long list of subatomic particles whose discovery initially baffled theoreticians who could not make

3080-852: Was elected to the United States National Academy of Sciences and the American Philosophical Society in 1938. He was elected a Fellow of the American Academy of Arts and Sciences in 1950. He received the Golden Plate Award of the American Academy of Achievement in 1975. He died on January 11, 1991, and his remains were interred in the Forest Lawn, Hollywood Hills Cemetery in Los Angeles, California . His wife Lorraine died in 1984. Hilbert%27s paradox of

3136-523: Was introduced by David Hilbert in a 1925 lecture " Über das Unendliche ", reprinted in ( Hilbert 2013 , p.730), and was popularized through George Gamow 's 1947 book One Two Three... Infinity . Hilbert imagines a hypothetical hotel with rooms numbered 1, 2, 3, and so on with no upper limit. This is called a countably infinite number of rooms. Initially every room is occupied, and yet new visitors arrive, each expecting their own room. A normal, finite hotel could not accommodate new guests once every room

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