Pascual Jordan's Resolution of the Conundrum of the Wave-Particle

Jordan’s resolution of the wave-particle conundrum of light

HQ1, Max Planck Institute for History of Science, Berlin, July 2–6, 2007 Pascual Jordan’s resolution of the conundrum of the wave-particle duality of light

Anthony Duncan Michel Janssen Department of Program in the History Physics and Astronomy, of Science, Technology, and Medicine University of Pittsburgh University of Minnesota

In 1909, Einstein derived a formula for the mean square energy fluctuation in a small subvolume of a box filled with black-body radiation. This formula is the sum of a wave term and a particle term. Einstein concluded that a satisfactory theory of light would thus have to combine aspects of a wave theory and a particle theory. In the following decade, various attempts were made to recover this formula without Einstein’s light quanta. In a key contribution to the 1925 Dreimännerarbeit with Born and Heisenberg, Jordan showed that a straightforward application of Heisenberg’s Umdeutung procedure, which forms the core of the new matrix mechanics, to a system of waves reproduces both terms of Einstein’s fluctuation formula. Jordan thus showed that these two terms do not require separate mechanisms, one involving particles and one involving waves. In matrix mechanics, both terms arise from a single consistent dynamical framework.

1 HQ1, MPIWG, Berlin, July 2-6, 2007

Jordan’s resolution of the wave-particle conundrum of light

Pascual Jordan’s resolution of the conundrum of the wave-particle duality of light

Outline

Einstein, ﬂuctuations, light quanta, and wave-particle duality

Resistance to the light-quantum hypothesis

Jordan’s resolution of the wave-particle conundrum in 3M

Jordan’s derivation of Einstein’s ﬂuctuation formula: the gory details

Jordan on the importance of his result

Conclusions

2 HQ1, MPIWG, Berlin, July 2-6, 2007 ! Jordan’s resolution of the wave-particle conundrum of light

Einstein, ﬂuctuations, light quanta, and wave-particle duality

1905: Einstein’s ‘ﬁrst ﬂuctuation formula’ (Jordan’s terminology) ! light quanta

V << V 0 V0 V0

V V

Entropy of black-body radiation in the Wien regime (high frequency) and Boltzmann’s principle S = klogW : E ! h" • Probability of ﬁnding all black-body radiation in V0 in subvolume V: "V ! V0# N • Probability of ﬁnding all N molecules of ideal gas in V0 in subvolume V: "V ! V0# • Einstein’s conclusion (“the miraculous argument” [Norton]): “radiation ... behaves ... as if it consisted of [N] mutually independent energy quanta of magnitude [h"].”

3 HQ1, MPIWG, Berlin, July 2-6, 2007 ! Jordan’s resolution of the wave-particle conundrum of light

1909: Einstein’s ‘second ﬂuctuation formula’ ! wave-particle duality

1904. Einstein applies formula for mean V << V 0 V0 square energy fluctuation (Boltzmann, Gibbs): d $E% V $#E2% = kT 2------dT to black-body radiation (final installment of the ‘statistical trilogy’; cf. Ryno-Renn, Norton, Uffink in Studies in History and Philosophy of Modern Physics 37 (2006) 1 [Centenary of Einstein’s Annus Mirabilis])

1909. Einstein’s GDNA lecture in Salzburg. Martin J. Klein:* “Instead of trying to derive the distribution law from some more fundamental starting point, he turned the argument around. Planck’s law had the solid backing of experiment; why not assume its correct- ness and see what conclusions it implied as to the structure of radiation?”

* “Einstein and the Wave-Particle Duality.” The Natural Philosopher 3 (1964): 1–49.

4 HQ1, MPIWG, Berlin, July 2-6, 2007

Jordan’s resolution of the wave-particle conundrum of light

V0 Task: ﬁnd the mean square ﬂuctuation in subvolume V of the V energy of black-body radiation at temperature T with spectral distribution &""$ T # in frequency range ""$ " + #"#.

Introduce: $E"% = &""$ T #V#", the average energy in frequency range ""$ " + #"# in subvolume V. V << V0 '&""$ T # Frequency-speciﬁc ﬂuctuation formula: $#E2% = kT 2------V#". " 'T Results for different black-body radiation laws: $E % 2 8( c3 " RJ Rayleigh-Jeans &RJ""$ T # = ------"2kT $#E2% = ------waves c3 " RJ 8("2 V#" 8(h Wien &W""$ T # = ------"3e–h" ! kT $#E2% = h" $E % particles c3 " W " W $E % 2 8(h "3 c3 " P Planck &P""$ T # = ------$#E2% = ------+ h" $E % c3 eh" ! kT – 1 " P 8("2 V#" " P waves + Einstein’s conclusion (1909): “the next phase of the development of particles! theoretical physics will bring us a theory of light that can be inter- preted as a kind of fusion of the wave and emission theories.”

5 HQ1, MPIWG, Berlin, July 2-6, 2007 Jordan’s resolution of the wave-particle conundrum of light

Resistance to the light-quantum hypothesis

• Planck and Nernst when recruiting Einstein for the Prussian Academy in Berlin (1914): “That he may sometimes have missed the target in his speculations, as for example, in his hypothesis of light quanta, cannot really be held against him.”

• Millikan after verifying Einstein’s predictions for the photoelectric effect (1916): “We are confronted by the astonishing situation that these facts were correctly and exactly predicted nine years ago by a form of quantum theory which has now been pretty generally abandoned.” (cf. Roger Stuewer in The Cambridge Companion to Einstein [in preparation])

• Mie to Wien, 6 February 1916: “It is as if his temperamental wishful thinking occasionally robs him of his capacity for calm deliberation, as once with his new theory of light, when he believed he could construct a light beam capable of interference out of nothing but incoherent parts.”

6 HQ1, MPIWG, Berlin, July 2-6, 2007 Jordan’s resolution of the wave-particle conundrum of light

Resistance to the light-quantum hypothesis persists even after the Compton effect.*

• Schizophrenic attitude of many physicists after Compton effect (1923): accept light quanta but continue to treat light as if it were a classical wave (particularly clear in dispersion theory: Ladenburg, Reiche, Van Vleck, Born, Heisenberg)

• Bohr’s last stand against light quanta: BKS—the Bohr-Kramers-Slater theory (1924). BKS “explains” Compton effect without light quanta.

• After Bothe & Geiger and Compton & Simon decisively refute BKS in 1925, Slater’s original idea of light quanta guided by a classical ﬁeld is resurrected (similar ideas: De Broglie, Einstein, Swann). Slater (in letter to Nature, July 25, 1925): “The simplest solution to the radiation problem then seems to be to return to the view of a virtual ﬁeld to guide corpuscular quanta.” Kramers to Urey, July 16, 1925: “We [here in Copenhagen] think that Slater’s original hypothesis contains a good deal of truth.”

*Duncan & Janssen. “On the Verge of Umdeutung in Minnesota: Van Vleck and the Correspondence Principle.” Archive for History of Exact Sciences (forthcoming).

7 HQ1, MPIWG, Berlin, July 2-6, 2007 Jordan’s resolution of the wave-particle conundrum of light

Attempts to recover Einstein’s ﬂuctuation formula without light quanta, 1909–1925

Albert Einstein and Ludwig Hopf, “Statistische Untersuchung der Bewegung eines Reso- nators in einem Strahlungsfeld.” AdP (= Annalen der Physik) 33 (1910): 1105–1115. Max von Laue, “Ein Satz der Wahrscheinlichkeitsrechnung und seine Anwendung auf die Strahlungstheorie.” AdP 47 (1915): 853–878. Albert Einstein, “Antwort auf eine Abhandlung M. v. Laues …” AdP 47 (1915): 879–885. Max von Laue, “Zur Statistik der Fourierkoefﬁzienten der natürlichen Strahlung.” AdP 48 (1915): 668–680. Max Planck, “Über die Natur der Wärmestrahlung.” AdP 73 (1924): 272–288. Paul Ehrenfest, “Energieschwankungen im Strahlungsfeld oder Kristallgitter bei Superposition quantisierter Eigenschwingungen.” ZfP (= Zeitschrift für Physik) 34 (1925): 362–373. Pascual Jordan, “Zur Theorie der Quantenstrahlung.” ZfP 30: (1924) 297–319. Albert Einstein, “Bemerkung zu P. Jordans Abhandlung …” ZfP 31 (1925): 784–785. Pascual Jordan, “Über das thermische Gleichgewicht zwischen Quantenatomen und Hohl- raumstrahlung.” ZfP 33 (1925): 649–655

8 HQ1, MPIWG, Berlin, July 2-6, 2007 Jordan’s resolution of the wave-particle conundrum of light

Jordan’s resolution of the wave-particle conundrum in 3M (= Dreimännerarbeit)

Max Born, Werner Heisenberg, Pascual Jordan, “Zur Quantenmechanik II.” Zeitschrift für Physik 35 (1925): 557–615. English translation in B. L. Van der Waerden (ed.), Sources of Quantum Mechanics. New York: Dover, 1968.

Einstein’s ﬂuctuation formula emerges from straightforward application of Heisen- berg’s Umdeutung procedure to classical waves. The formula does not require separate mechanisms, one involving particles and one involving waves. In matrix mechanics, both terms arise from a single consistent dynamical framework.

Presented in 3M but actually Jordan’s work:

Heisenberg to Pauli, October 23, 1925: “Jordan claims that the interference calculations come out right, both the classical and the Einsteinian terms … I am rather sorry that I do not understand enough statistics to be able to judge how much sense it makes; but I cannot criticize either, because the problem itself and the subsequent calculations sound meaningful.”

Jordan to Van der Waerden, April 10, 1962: “[my] most important contribution to quantum mechanics.”

9 HQ1, MPIWG, Berlin, July 2-6, 2007 Jordan’s resolution of the wave-particle conundrum of light

Other works by Jordan on ﬂuctuations in radiation and on ﬁeld quantization* 1924. “Zur Theorie der Quantenstrahlung.” ZfP 30: 297–319. Einstein’s response: “Bemerkung zu P. Jordans Abhandlung …” ZfP 31: 784–785. 1925. “Über das thermische Gleichgewicht zwischen Quantenatomen und Hohlraumstrahlung.” ZfP 33: 649–655 1927. “Über Wellen und Korpuskeln in der Quanten- mechanik.” ZfP 45: 766–775. 1927 (with Oskar Klein), “Zum Mehrkörperproblem der Quantentheorie.” ZfP 45: 751–765. 1928 (with Eugene Wigner), “Über das Paulische Äquivalenzverbot.” ZfP 47: 631–651. 1928. “Die Lichtquantenhypothese. Entwicklung und gegenwärtiger Stand.” Ergebnisse der exakten Naturwissenschaften 7: 158–208. 1930 (with Max Born), Elementare Quanten- Pascual Jordan (1902–1980) mechanik. Berlin: Springer. *Cf. Olivier Darrigol, “The origin of quantized matter waves.” Historical Studies in the Physical and Biological Sciences 16 (1986): 197–253.

10 HQ1, MPIWG, Berlin, July 2-6, 2007 Jordan’s resolution of the wave-particle conundrum of light

Jordan’s derivation of Einstein’s ﬂuctuation formula: the gory details

Classical model (Ehrenfest): String of length l ﬁxed at both ends (= 1D analogue of EM ﬁeld in box with conducting sides). Displacement u"x$ t#

1 l Hamiltonian: H = -- dx"u˙2 + u2# 2) x 0 + Insert Fourier expansion u"x$ t# = , qk"t#sin"*kx# with Fourier coefﬁcients k = 1 ( q "t# = a cos"* t + - # and frequencies * 2 k0--1 : k k k k k . l/ 1 l + H = -- dx "q˙ "t#q˙ "t#sin"* x#sin"* x# + * * q "t#q "t#cos"* x#cos"* x## 2 ) , j k j k j k j k j k 0 j$ k = 1 l l Only contributions for j = k: %sin"* x#& orthogonal on "0$ l#: dxsin* xsin* x = --3 j ) j k 2 jk 0 H turns into Hamiltonian for inﬁnite set of uncoupled oscillators of mass l/2: + + l H = --"q˙2"t# + *2q2"t## = H , 4 j j j , j j = 1 j = 1

11 HQ1, MPIWG, Berlin, July 2-6, 2007 Jordan’s resolution of the wave-particle conundrum of light

Hamiltonian for string turns into Hamiltonian for inﬁnite set of uncoupled oscillators. l 1 + l H = -- dx"u˙2 + u2# = H with H = --"q˙2"t# + *2q2"t## 2 ) x , j j 4 j j j 0 j = 1 • Distribution of energy over frequencies is constant in time. • Distribution of energy in some frequency range over length of string varies in time.

Task: calculate the mean square ﬂuctuation of the energy in a narrow frequency range "*$ * + #*# in a small segment "0$ a# of the string (a << l ). small segment "0$ a# of the string Two steps 1. (Jordan’s version of) Ehrenfest’s classical calculation 2. Jordan’s quantum-mechanical version of the calculation

12 HQ1, MPIWG, Berlin, July 2-6, 2007 Jordan’s resolution of the wave-particle conundrum of light

1. Classical calculation of mean square ﬂuctuation of energy in "*$ * + #*# in "0$ a# Instantaneous energy in "*$ * + #*# in "0$ a# a 1 E "t# = -- dx %q˙ "t#q˙ "t#sin"* x#sin"* x# "*$ a# 2 ) , j k j k 0 * ' j"( ! l# ' * + #*

* ' k"( ! l# ' * + #* !+! !*! j!*! k! q! j"t#qk"t#cos"* jx#cos"*kx# &

%sin"* jx#& not orthogonal on "0$ a# ! Both " j = k#-terms and " j 4 k#-terms contribute. • " j = k#-terms: average energy in "*$ * + #*# in "0$ a#: a a • E" j = k# t = -- q˙2 t + 2q2 t = -- H "*$ a# " # ," j " # * j j " ## , j 4 j l j ! ! ! ! • H ’s are constant E" j = k# t = E" j = k# t (bar means time average) j !! "*$ a# " # "a$ *# " #

• For j 4 k, q˙ "t#q˙ "t# = q "t#q "t# = 0 ! E" j 4 k#"t# = 0 j k j k "*$ a# E" j = k# t = E t = fraction "a ! l#of the (constant) total amount of ! "*$ a# " # "*$ a#" # energy in this frequency range in the entire string • " j 4 k#-terms: instantaneous ﬂuctuation of energy in "*$ * + #*# in "0$ a#: E" j 4 k# t = E t – E t = #E t ! "*$ a#" # "*$ a#" # "*$ a#" # # "*$ a#" #

13 HQ1, MPIWG, Berlin, July 2-6, 2007 Jordan’s resolution of the wave-particle conundrum of light

Instantaneous energy ﬂuctuation in "*$ * + #*# in "0$ a#

E t = E t – E t = E" j 4 k# t # "*$ a#" # "*$ a#" # "*$ a#" # "*$ a# " #

a ! = !) !d x! ! ,! ! !" q!˙ !j q!˙! k !( cos""* j – *k#x# – cos""* j + *k#x#) 0 j 4 k !+ ! * j! *! k! q! j!q ! k !( !cos ! " " *! !j –! !* ! k! # x! #! +! !cos ! ! "! "!!* j + *k#x#) #

2 ! K jk 1 0 sin""* j – *k#a# sin""* j + *k#a# = -- , 5q˙ jq˙k ------– ------2 ! K 7 4 j 4 k. * j – *k * j + *k jk

sin""* j – *k#a# sin""* j + *k#a# 1 !+ ! * j! *! k! q! j!q ! k ! --!- - --!- --!- --!- --!- -- -!------+ ------6 * j – *k * j + *k /

Result (suppress time dependence): 1 #E = -- q˙ q˙ K + * * q q K 7 # "*$ a# , " j k jk * j*k j k jk # 4 j 4 k

14 HQ1, MPIWG, Berlin, July 2-6, 2007 Jordan’s resolution of the wave-particle conundrum of light

Energy ﬂuctuation in range "*$ * + #*# in segment "0$ a#: Instantaneous fluctuation 1 #E = -- "q˙ q˙ K + * * q q K 7 # 2 #E + #E "*$ a# 4 , j k jk j k j k jk 1 2 j 4 k "*$ a# "*$ a# Mean square fluctuation

E2 = E2 + E2 + E E + E E # *$ a # 1 # 2 # 1 # 2 # 2 # 1 "*$ # "*$ a# "*$ a# "*$ a# "*$ a# "*$ a# "*$ a# • Square terms

#E2 + #E2 = ! ! ! ! ! !! ! ! !! ! ! ! ! ! ! !! !!! ! ! ! ! ! ! ! ! 1 a 2 a "*$ # 1 "*$ # ----- 7 7 , , "q˙ jq˙kq˙ j7q˙k7K jkK j7k7 + q jqkq j7qk7* j*k* j7*k7K jk K j7k7 # 16 j 4 k j7 4 k7 • Cross terms

#E1 #E2 + #E2 #E1 = ! !! !! ! ! ! ! !! ! ! ! ! ! ! !! !! "*$ a# "*$ a# "*$ a# "*$ a# 1 ----- 7 7 , , "q˙ jq˙kq j7qk7* j7*k7K jkK j7k7 + q jqkq˙ j7q˙k7* j*kK jk K j7k7# 16 j 4 k j7 4 k7 Classically: only square terms contribute to E2 . Quantum-mechanically: square # "*$ a# terms and cross terms contribute.

15 HQ1, MPIWG, Berlin, July 2-6, 2007 Jordan’s resolution of the wave-particle conundrum of light

Evaluation of time averages of products of q’s and q˙’s in square terms of E2 # "*$ a# #E2 + #E2 = ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! !! ! !! ! ! ! ! ! ! ! ! ! ! ! !! ! ! 1 a 2 a "*$ # 1 "*$ # ----- q˙ q˙ q˙ q˙ K K + q q q q K 7 K 7 , , " j k j7 k7 jk j7k7 j k j7 k7* j*k* j7*k7 jk j7k7# 16 j 4 k j7 4 k7

• Recall qk"t# = ak cos"*kt + -k#. • Argument for time averages qk"t#$ q˙k"t# also works for phase averages (qk"t#)$ (q˙k"t#) (square brackets: average over -k )

q jqkq j7qk7 8 cos"* jt#cos"*kt#cos"* j7t#cos"*k7t# = 0 unless " j = j7$ k = k7# or " j = k7$ k = j7# q˙ jq˙kq˙ j7q˙k7 8 sin"* jt#sin"*kt#sin"* j7t#sin"*k7t#

Cross terms (2 cosines, 2 sines):

q jqkq˙ j7q˙k7 8 cos"* jt#cos"*kt#sin"* j7t#sin"*k7t# = 0 for all index combinations

1 #E2 = #9+ , #9+ = -- 0q˙2q˙2K 2 + q2q2*2*2K 721 "*$ a# * + 8 , . j k jk j k j k jk/ "*$ :# "*$ :# j 4 k

16 HQ1, MPIWG, Berlin, July 2-6, 2007 Jordan’s resolution of the wave-particle conundrum of light

Mean square energy ﬂuctuation in range "*$ * + #*# in segment "0$ a#: 1 E2 = -- q˙2q˙2K 2 + q2q2 2 2K 72 # "*$ a# , " j k jk j k * j *k jk # 8 j 4 k 1 l l 1 l Virial theorem for oscillator, E = E = --E : --q˙2 = --*2q2 = --H 0m = --1 -! kin pot 2 tot 4 j 4 j j 2 j . 2/ 2 2 Hence,!q˙2 = --H and q2 = ------H : j l j j 2 j l* j 1 #E2 = ------H H "K 2 + K 72 # "*$ a# 2 , j k jk jk 2l j 4 k From discrete sums to integrals (requires extra assumption: distribution of energy over l ( frequency is smooth, i.e., H varies smoothly with j): , ! ! !- -!! d* 0since !* ! != j--1 , j j () . j l/ l * ,! ! ! !- -!! d*7, H H ! H H , K 2 ! K 2 ! (a3"* – *7#: k () j k * *7 jk **7

1 l 2 a #E2 = ------d* d*70--1 2(a3"* – *7#H H = -- d*H2 "*$ a# 2l2) ) .(/ * *7 () *

2 * 2 sin ""* – *7#a# )d*7 f "*7#K = )d*7 f "*7#------= )d*7 f "*7#(a3"* – *7# = (af "*# **7 "* – *7#2

17 HQ1, MPIWG, Berlin, July 2-6, 2007 Jordan’s resolution of the wave-particle conundrum of light

Mean square energy ﬂuctuation in segment "0$ a# …

… in range "*$ * + #*#: … in range ""$ " + #"#: a E2 = -- d H 2 f "*# ! f ""# # a ) * * 2 2 "*$ # ( #E""$ a# = 2a#"H" d 2 ) * ! ! .! ! !!(#! "! ! ! ! ! ! ! ! ! ! 2 E""$ a# a ! !! =! ! --!- --!------E = --H N 2a#" ""$ a# l " " N = number of modes in + Mean square energy " ""$ " #"# ﬂuctuations proportional ( to mean energy squared. N -- = #* = 2(#" ! N = 2l#" " l " Result for classical waves. E = 2a#"H "a$ "# "

Comment: This is a very general result (already found by Lorentz in 1912/16). It holds for any smooth distribution of energy over frequency (equilibrium or not). Temperature does not enter into the derivation at all.

18 HQ1, MPIWG, Berlin, July 2-6, 2007 Jordan’s resolution of the wave-particle conundrum of light

2. Quantum calculation of the mean square ﬂuctuation of the energy in a narrow frequency range "*$ * + #*# in a small segment "0$ a# of the string.

Preview: All quantities—in modern terms—become operators. Because operators q and ! ! q˙ don’t commute, the quantum analogue of the cross terms #E1#E2 + #E2#E1 con- tributes to the mean square energy ﬂuctuation. This contribution is essential for correctly reproducing the particle term in the Einstein ﬂuctuation formula.

Key point of Heisenberg’s Umdeutung paper: operators representing position and

momentum still satisfy the old equations of motion ! Operator qk"t# for quantum 2 harmonic oscillator satisﬁes classical equation q˙˙k"t# = –*k qk"t#. Solution:

2 pk"0# qk"t# = qk"0#cos"*kt# + ------sin"*kt# l*k 2 pk"0# q˙ "t# = ------cos"* t# – * q "0#sin"* t# k l k k k k

19 HQ1, MPIWG, Berlin, July 2-6, 2007 Jordan’s resolution of the wave-particle conundrum of light

Operator for mean square energy ﬂuctuation in range "*$ * + #*# in segment "0$ a#: 2 2 2 #H a = #H1 + #H2 + #H1 #H2 + #H2 #H1 "*$ # "*$ a# "*$ a# "*$ a# "*$ a# "*$ a# "*$ a# Consider cross terms (which vanished classically):

#H1 #H2 + #H2 #H1 "*$ a# "*$ a# "*$ a# "*$ a# 1 ----- 7 7 = , , "q˙ jq˙kq j7qk7* j7*k7K jkK j7k7 + q jqkq˙ j7q˙k7* j*kK jk K j7k7# 16 j 4 k j7 4 k7 1 No longer vanishes = -- "q˙ q q˙ q + q q˙ q q˙ #* * K K 7 , j j k k j j k k j k jk jk ˙ 8 j 4 k since q j and q j do not commute! Evaluate q˙kqk:

2 pk"0# 0 2 pk"0# 1 q˙ q = 0------cos"* t# – * q "0#sin"* t#1 q "0#cos"* t# + ------sin"* t# k k . k k k k / 5 k k k 6 l . l*k / 2 1 ih = --" p "0#q "0# – q "0# p "0##-- = –----- l k k k k 2 l Hence: 1 h2 #H #H + #H #H = –------* * K K 7 1 2 2 1 2 , 4 j k jk jk "*$ a# "*$ a# "*$ a# "*$ a# l j 4 k

20 HQ1, MPIWG, Berlin, July 2-6, 2007 Jordan’s resolution of the wave-particle conundrum of light

Operator for mean square energy ﬂuctuations in range "*$ * + #*# in segment "0$ a#: Contribution from cross terms (because of non-commutativity of p and q): 1 h2 #H #H + #H #H = –------* * K 2 1 2 2 1 2 , 4 j k jk "*$ a# "*$ a# "*$ a# "*$ a# l j 4 k Contribution from square terms (same as before) Note: K 2 , K7 2 , and K K7 1 jk jk jk jk #H 2 + #H 2 = ---- H H K 2 1 2 2 , j k jk can be used interchangeably in "*$ a# "*$ a# l j 4 k these sums Added together 1 h2 #H2 = ---- 0H H – ------* * 1 K2 "*$ a# 2 , . j k j k/ jk l j 4 k 4

From discrete sums to integrals (on the assumption that H j varies smoothly with j): l 2 , !! ! !0 !-- !1! !! d* d*7, H H ! H H , K 2 ! K 2 ! (a3"* – *7#: j 4 k .(/ ) ) j k * *7 jk "**7# 1 l 2 h2 a h* 2 #H2 = ---- d* d*70--1 0H H – ------**71 (a3"* – *7# = -- d*0H2 – 0------1 1 "*$ a# l2) ) .(/ . * *7 16(2 / () . * . 2 / /

21 HQ1, MPIWG, Berlin, July 2-6, 2007 Jordan’s resolution of the wave-particle conundrum of light

Operator for mean square energy ﬂuctuations in segment "0$ a# … … in range "*$ * + #*#: … in range ""$ " + #"#: a h* 2 h" 2 H 2 = -- d 0H 2 – 0------1 1 #H2 = 2a#"0H2 – 0------1 1 # "*$ a# ) * * ""$ a# " ( . . 2 / / f "*# ! f ""# . . 2 / / )d* ! ! .! 2(#" Expectation value of mean square energy ﬂuctuation in ""$ " + #"# in "0$ a#

(1) 1 2 1 #E2 2 $%n &/#H 2 /%n &% = 2a#"00n + --1 – --1 h2"2 ""$ a# " ""$ a# " .. " 2/ 4/ (2) 2 ! ! ! ! ! ! ! = 2a#"""n"h"# + "n"h"#h"# 1 h" 0 1 ------2 (1) $%n &/H /%n &% = n" + -- h" with the zero- E " " " . 2/ 2 (3) ""$ a# ! ! !! ! ! !! ! = ------+ E h" point energy 2a#" ""$ a# nd (2) 2 term, coming from ( p$ q) 4 0, cancels the square Exactly the same form as the of the zero-point energy coming from 1st term. Einstein ﬂuctuation formula (3) ‘Renormalized’ energy (with zero-point energy sub- with wave and particle term! a tracted): E 2 --"n h"#N = 2a#""n h"# ""$ a# l " " "

Number of modes in in ""$ " + #"#: N" = 2l#"

22 HQ1, MPIWG, Berlin, July 2-6, 2007 Jordan’s resolution of the wave-particle conundrum of light

Expectation value of the mean square ﬂuctuation of the energy in ""$ " + #"# in "0$ a# E 2 ""$ a# #E2 = ------+ E h" ""$ a# 2a#" ""$ a#

Comments: • Since the entire derivation is for a ﬁnite frequency range, there are no problems with inﬁnities (pace, e.g., Jürgen Ehlers at HQ0, Mehra-Rechenberg, Vol. 3, p. 156). • Like its classical counterpart, this quantum formula is a very general result. It holds for all states %n"& in which n" is a reasonably smooth function of ". There is no restric- tion to equilibrium states. Again, temperature does not enter into the derivation.

• The formula is for the quantum spread of the energy in a narrow frequency range in a small segment of the string in an energy eigenstate of the whole system. The operators H and H"*$ a# do not commute. The system is in an eigenstate of the full Hamiltonian H but in a superposition of eigenstates of H"*$ a#. • To get from the formula for the quantum spread to a formula for thermodynamical ﬂuctuations, we need to go from energy eigenstates to thermal averages of such states. It can be shown that this transition does not affect the form of the formula (key point: formula only involves products of thermally independent modes).

23 HQ1, MPIWG, Berlin, July 2-6, 2007 Jordan’s resolution of the wave-particle conundrum of light

Claim: The formula for the thermodynamical ﬂuctuation of the energy in a narrow frequency range in a small subvolume has the exact same form as the formula for the quantum spread in that energy. –;E %ni& , $%ni&/O/%ni&%e %n & Thermal average for operator O: ($%n &/O/%n &%) = ------i------i i –;E %n & • Square brackets for canonical , e i ensemble expectation value %ni& 1 • ; = ------Boltzmann factor kT 1 • E = 0n + --1 h" %n & ,. i / i i i 2

2 Naively: n H 2 n $%n &/#H /%n &% $% "&/# ""$ a#/% "&% " ""$ a# " quantum spread thermal average E 2 E 2 ""$ a# ( ""$ a# ) #E2 = ------+ E h" #E2 = ------+ (E )h" ""$ a# 2a#" ""$ a# ""$ a# 2a#" ""$ a#

Wrong result: E 2 should be E 2! ( ""$ a# ) ( ""$ a#)

24 HQ1, MPIWG, Berlin, July 2-6, 2007 Jordan’s resolution of the wave-particle conundrum of light

1 h2 Calculate $%n &/#H 2 /%n &% with #H 2 = ---- 0H H – ----- " " 1 K 2 i ""$ a# i ""$ a# 2 , . j k 4 j k/ jk l j 4 k 1 1 Only non-trivial part: –;00n + --1 h" + 0n + --1 h" 1 1 1 .. j 2/ j . k 2/ k/ 0n + --1 h" 0n + --1 h" e , , . j / j. k / k n n 2 2 ($%n &/H H /%n &%) = ----j-----k------i j k i 1 1 –;00n + --1 h" + 0n + --1 h" 1 .. j 2/ j . k 2/ k/ , , e n n j k Product of factor for jth mode and factor for thermally independent kth mode: 1 –;0n + --1 h" 1 . j 2/ j ,0n + --1 h" e . j 2/ j 1 1 ------= 0nˆ + --1 h" with: nˆ 2 ------(Planck function 1 . j 2/ j j ;h" –;0n + --1 h" e j – 1 for average ther- . j 2/ j ,e mal excitation th Insert: level of j mode) 1 1 $%n &/#H 2 /%n &% = ---- 0nˆ nˆ + --"nˆ + nˆ #1 h2" " K 2 i ""$ a# i 2 , . j k 2 k k / j k jk l j 4 k

25 HQ1, MPIWG, Berlin, July 2-6, 2007 Jordan’s resolution of the wave-particle conundrum of light

Claim: The formula for the thermodynamical ﬂuctuation of the energy in a narrow frequency range in a small subvolume has the exact same form as the formula for the quantum spread in that energy.

Formula for discrete sums: 1 1 $%n &/#H 2 /%n &% = ---- 0nˆ nˆ + --"nˆ + nˆ #1 h2" " K 2 i ""$ a# i 2 , . j k 2 k k / j k jk l j 4 k From discrete sums to integrals (no extra assumption needed: nˆ j varies smoothly with " j)

(#E2 ) 2 $%n &/#H 2 /%n &% = 2a#""nˆ 2 + nˆ #h2"2 ""$ a# " ""$ a# " " "

Use (E""$ a#) = 2a#""nˆ "h"#: 2 (E""$ a#) (#E2 ) = ------+ (E )h" thermodynamical ﬂuctuation ""$ a# 2a#" ""$ a# E 2 ""$ a# #E2 = ------+ E h" quantum spread ""$ a# 2a#" ""$ a# Two formulae have exactly the same structure.

26 HQ1, MPIWG, Berlin, July 2-6, 2007 Jordan’s resolution of the wave-particle conundrum of light

Jordan on the importance of his result

Conclusion of 3M: “If one bears in mind that the question considered here is actually somewhat remote from the problems whose investigation led to the growth of quantum mechanics, the result … can be regarded as particularly encouraging for the further development of the theory.” “The basic difference between the theory proposed here and that used hitherto … lies in the characteristic kinematics and not in a disparity of the mechanical laws. One could indeed perceive one of the most evident examples of the difference between quantum- theoretical kinematics and that existing hitherto on examining [the quantum ﬂuctuation formula], which actually involves no mechanical principles whatsoever [sic].”

27 HQ1, MPIWG, Berlin, July 2-6, 2007 Jordan’s resolution of the wave-particle conundrum of light

Jordan on the importance of his result

From Jordan’s 1928 history of the light-quantum hypothesis: “it turned out to be superﬂu- ous to explicitly adopt the light-quantum hypothesis: We explicitly stuck to the wave theory of light and only changed the kinematics of cavity waves quantum-mechanically. From this, however, the characteristic light-quantum effects emerged automatically as a consequence … This is a whole new turn in the light-quantum problem. It is not necessary to include the picture [Vorstellung] of light quanta among the assumptions of the theory. One can—and this seems to be the natural way to proceed—start from the wave picture. If one formulates this with the concepts of quantum mechanics, then the characteristic light-quantum effects emerge as necessary consequences from the general laws [Gesetzmäßigkeiten] of quantum theory.”

From the 1930 Born-Jordan book on matrix mechanics: “Consideration of the Wien limit of the Planck radiation law suggests, according to Einstein, that the wave picture has to be replaced or supplemented by the particle picture. Quantum mechanics, however, makes it possible to restore agreement with Planck's law and all its consequences without giving up the wave picture. It sufﬁces to work through the model [Modellvorstel- lung] of the classical wave theory with the exact quantum-theoretical kinematics and mechanics. The characteristic light-quantum effects then emerge automatically, without the addition of new hypotheses, as necessary consequences of the wave theory.”

28 HQ1, MPIWG, Berlin, July 2-6, 2007 Jordan’s resolution of the wave-particle conundrum of light

Conclusions

Jordan’s derivation of Einstein’s ﬂuctuation formula in 3M

The derivation • is tricky but kosher.

The result • resolves the conundrum of the wave-particle duality for light; • was not hailed as a major breakthrough in its own right in 3M (and Jordan’s subsequent expositions did not get much attention); • can be seen as the birth of quantum ﬁeld theory.

29 HQ1, MPIWG, Berlin, July 2-6, 2007