Carrying a Polarisation Around a Closed Path Shifts Its Phase by Half the Enclosed Solid Angle ── enclosing a hemisphere gives a phase of pi and flips the sign, the same half as the Berry phase of a qubit ── the separator is whether one measures intensity alone or the phase by interference with a reference ── [Paper 906]
Passing light through a few polarisers and back to its original polarisation, it is easy to assume the light has returned to what it was. This paper follows the state of polarisation as a point on the Poincaré sphere and compares the phase left after a closed path with the solid angle the path encloses. No new theorem or law is claimed. Scope of this paper (scope note): No new theorem or law is claimed──that a closed path of polarisations acquires a phase of half the solid angle (Pancharatnam 1956), that this is the same as Berry’s phase (1984; Berry 1987), and that the state space of polarisation is a Hopf fibration are all known. No real optical system is claimed──the number, angles and amplitudes of the polarisers are model values, not those of any apparatus. Partial polarisation and loss are not entered──only projections of fully polarised light are treated; partial polarisation, differential absorption and non-adiabatic effects are not included. The origin of the phase is not entered──that the phase is the holonomy of a Hopf fibration is the matter of Paper 68; here only the numbers for light are given. Relation to earlier papers: Paper 68 showed that the qubit is a Hopf fibration and that the Berry phase is measurable as the holonomy -Omega/2 of its fibre──the two-component Jones vector used here has the same form as a spin-1/2 spinor, and the same function gives the same phase. Since the formula, not merely the number, is the same, this is written as the same root in the sense of Paper 300. Paper 207 showed that there are three tools for writing polarisation and that the Jones vector cannot write unpolarised light──here the closed path is followed on the third of them, the Poincaré sphere. Paper 08 showed that in the Hopf fibration (the map from S^3 to S^2 whose fibre is S^1) the area of the base times the circumference of the fibre is the volume of the whole──here the base is the Poincaré sphere and the fibre is the phase of the polarisation. Paper 18 separated three roots of 2pi and took one of them to be the solid angle of a hemisphere──here the hemisphere’s solid angle 2pi becomes a phase of pi. What is added is checking the phase of closed paths made of polariser projections against -Omega/2 on random triangles, showing in numbers that on the hemispherical path the amplitude returns while the sign does not, measuring the convergence as the steps are refined, and placing the separator on intensity versus interference. First, the map to the state of polarisation is the Hopf map──the Stokes vectors built from 2000 two-component complex vectors all lie on the unit sphere and do not move, to the order of 10^-15, when an overall phase is applied (Section 2). Second, and this is the core. The phase of a closed path is half the solid angle enclosed──returning through three polarisers A-> B-> C-> A, the phase arg( A|C)( C|B)( B|A) agreed with -Omega/2 of the enclosed solid angle within 10^-9 for five random triangles (for instance Omega=-5.580641 gives phase +2.790320). The reverse path gives +Omega/2, which equals -(4pi-Omega)/2 modulo 2pi (Section 2). Third, enclosing a hemisphere flips the sign──turning a linear polarisation through half a turn in N steps and back (once around the equator, Omega=2pi) gives a phase of pi for every N from 3 to 384 (Section 3). Fourth, the amplitude returns but the sign does not──the surviving amplitude is cos^N(pi/N), rising from 0.125000 at N=3 to 0.987231 at N=384, yet the intensity when overlaid in phase with a reference cancels to 0.000041 at N=384 (Section 3). Fifth, finer steps converge to the half on any path──around a circle of angular radius 10 degrees near circular polarisation (Omega=0.095456), the phase moves from -0.039621 at N=6 to -0.047728 at N=1536, within 1.30x10^-7 of -Omega/2 (Section 4). Sixth, the separator is whether one measures intensity alone or the phase by interference with a reference──by intensity alone, the finer the steps the more fully the light returns; overlaid with a reference, the hemisphere’s phase pi remains as darkness. The formula is that of the same two-component vector as a spin-1/2 qubit, and shares its root with the -Omega/2 of Paper 68 (Section 5). Carrying a polarisation around a closed path shifts its phase by half the solid angle enclosed, and enclosing a hemisphere flips the sign: the same half as the Berry phase of a qubit. The phases of triangular paths made of polariser projections agreed with -Omega/2 in all five cases, and on the path that turns a linear polarisation through half a turn and back the phase was pi independently of N──the surviving amplitude returned to 0.987231 at N=384, yet overlaid with a reference the intensity cancelled to 0.000041. On a small circle too the phase converged to -Omega/2 as the steps were refined, the reverse path gave +Omega/2, and the full sphere 4pi corresponded to a phase of 2pi──what returned was the state; the phase remembers half the sky enclosed. Placed among the earlier papers──the same formula as the qubit of Paper 68 (the same root) was counted with light, on the Hopf fibration of Paper 08 and the Poincaré sphere of Paper 207. To be honest──the apparatus values are a model, neither partial polarisation nor loss was included, and the origin of the phase was left to Paper 68. On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated. Keywords: polarisation, Poincaré sphere, geometric phase, solid angle, Hopf map, interference. ----- 偏光子を何枚か通して元の偏光に戻せば、光は元に戻った、と思いやすい。本稿は、偏光の状態をポアンカレ球の点として追い、閉じた道を一周したあとに残る位相を、囲んだ立体角と比べる。新しい定理も法則も主張しない。 本稿の射程(射程注記):新しい定理も法則も主張しない──偏光の閉じた道に立体角の半分の位相が付くこと(パンチャラトナム 1956)、それがベリー位相(1984)と同じであること(ベリー 1987)、偏光の状態空間がホップ束であることは、いずれも既知である。実在の光学系を主張しない──偏光子の枚数・角度・振幅は模型の値で、どの装置の値でもない。部分偏光と損失を入れない──完全偏光の射影だけを扱い、部分偏光、吸収の違い、非断熱の効果は入れない。位相の由来に入らない──位相がホップ束のホロノミーであることの説明は論文68 の側で、ここでは光での数だけを出す。既刊との関係:論文68 は、量子ビットがホップ束であり、ベリー位相がその繊維のホロノミー -Omega/2 として測れることを示した──ここで使う二成分のジョーンズ・ベクトルはスピン 1/2 のスピノルと同じ形で、同じ関数から同じ位相が出る。数の一致ではなく式が同じなので、論文300 の意味で同根と書く。論文207 は、偏光を書く道具が三つあり、ジョーンズ・ベクトルは偏光していない光を書けないことを示した──ここではその三つ目、ポアンカレ球の上で閉じた道を追う。論文08 は、ホップ束(S^3 から S^2 への写像で、繊維は S^1)で底空間の面積と繊維の周長の積が全空間の体積になることを示した──ここでは、その底空間がポアンカレ球、繊維が偏光の位相である。論文18 は、2pi の三つの別根を分け、その一つを半球の立体角とした──ここでは半球の立体角 2pi が、位相の pi になる。加えたのは、偏光子の射影で作った閉じた道の位相を無作為の三角形で -Omega/2 と突き合わせたこと、半球の道で振幅が戻っても符号が戻らないことを数で示したこと、段を細かくしたときの収束を測ったこと、分離子を「強度か、干渉か」に置いたことである。 第一に、偏光の状態への写像はホップ写像である──二成分の複素ベクトルから作ったストークス・ベクトルは 2000 個すべてで単位球に乗り、全体の位相を掛けても 10^-15 の桁で動かない(第2節)。 第二に、これが本稿の芯である。閉じた道の位相は、囲んだ立体角の半分である──偏光子三枚で A-> B-> C-> A と戻したときの位相 arg( A|C)( C|B)( B|A) は、五つの無作為の三角形で囲んだ立体角の -Omega/2 と 10^-9 以内で一致した(例:Omega=-5.580641 で位相 +2.790320)。逆回りは +Omega/2 で、-(4pi-Omega)/2 と 2pi を法として同じである(第2節)。 第三に、半球を囲めば、符号が反転する──直線偏光の向きを N 段で半回転させて戻す(赤道を一周・Omega=2pi)と、位相は N=3 から 384 のすべてで pi である(第3節)。 第四に、振幅は戻るのに、符号は戻らない──残る振幅は cos^N(pi/N) で、N=3 の 0.125000 から N=384 の 0.987231 まで 1 に近づくのに、参照光と同位相で重ねた強度は N=384 で 0.000041 に打ち消す(第3節)。 第五に、段を細かくすると、どの道でも半分に収束する──円偏光の近くの角半径 10 度の円(Omega=0.095456)を N 段で一周すると、位相は N=6 の -0.039621 から N=1536 の -0.047728 へ、-Omega/2 に 1.30x10^-7 まで近づく(第4節)。 第六に、分離子は、強度だけを測るか、参照と干渉させて位相を測るかである──強度だけなら N を増やすほど元に戻り、参照と重ねれば半球の位相 pi が暗さとして残る。式はスピン 1/2 の量子ビットと同じ二成分ベクトルのもので、論文68 の -Omega/2 と同根である(第5節)。 偏光を閉じた道で一周させると位相は囲んだ立体角の半分だけ動き、半球を囲めば符号が反転する──量子ビットのベリー位相と同じ半分である。偏光子の射影で作った三角形の道の位相は五つとも -Omega/2 と一致し、直線偏光の向きを半回転させて戻す道では位相が N に依らず pi だった──残る振幅は N=384 で 0.987231 まで戻るのに、参照光と重ねた強度は 0.000041 に打ち消す。小さい円でも段を細かくすれば -Omega/2 に収束し、逆回りは +Omega/2 で、全球 4pi が位相の 2pi に対応する──元に戻ったのは状態で、位相は囲んだ空の半分を覚えている。既刊との位置──論文68 の量子ビットと同じ式(同根)を光で数え、論文08 のホップ束と論文207 のポアンカレ球の上に置いた。正直に言えば──装置の値は模型で、部分偏光も損失も入れず、位相の由来は論文68 に任せた。 作成にあたって:本稿の着想と内容は、著者自身の考察に基づくものです。文章の構成整理や英訳、数式の確認には AI(大規模言語モデル)の助力を得ました。最終的な内容の解釈や誤りがあれば、それらはすべて著者の責に帰します。お気づきの点があれば、ご教示いただければ幸いです。 キーワード:偏光、ポアンカレ球、幾何学的位相、立体角、ホップ写像、干渉。
Passing light through a few polarisers and back to its original polarisation, it is easy to assume the light has returned to what it was. This paper follows the state of polarisation as a point on the Poincaré sphere and compares the phase left after a closed path with the solid angle the path encloses. No new theorem or law is claimed. Scope of this paper (scope note): No new theorem or law is claimed──that a closed path of polarisations acquires a phase of half the solid angle (Pancharatnam 1956), that this is the same as Berry’s phase (1984; Berry 1987), and that the state space of polarisation is a Hopf fibration are all known. No real optical system is claimed──the number, angles and amplitudes of the polarisers are model values, not those of any apparatus. Partial polarisation and loss are not entered──only projections of fully polarised light are treated; partial polarisation, differential absorption and non-adiabatic effects are not included. The origin of the phase is not entered──that the phase is the holonomy of a Hopf fibration is the matter of Paper 68; here only the numbers for light are given. Relation to earlier papers: Paper 68 showed that the qubit is a Hopf fibration and that the Berry phase is measurable as the holonomy -Omega/2 of its fibre──the two-component Jones vector used here has the same form as a spin-1/2 spinor, and the same function gives the same phase. Since the formula, not merely the number, is the same, this is written as the same root in the sense of Paper 300. Paper 207 showed that there are three tools for writing polarisation and that the Jones vector cannot write unpolarised light──here the closed path is followed on the third of them, the Poincaré sphere. Paper 08 showed that in the Hopf fibration (the map from S^3 to S^2 whose fibre is S^1) the area of the base times the circumference of the fibre is the volume of the whole──here the base is the Poincaré sphere and the fibre is the phase of the polarisation. Paper 18 separated three roots of 2pi and took one of them to be the solid angle of a hemisphere──here the hemisphere’s solid angle 2pi becomes a phase of pi. What is added is checking the phase of closed paths made of polariser projections against -Omega/2 on random triangles, showing in numbers that on the hemispherical path the amplitude returns while the sign does not, measuring the convergence as the steps are refined, and placing the separator on intensity versus interference. First, the map to the state of polarisation is the Hopf map──the Stokes vectors built from 2000 two-component complex vectors all lie on the unit sphere and do not move, to the order of 10^-15, when an overall phase is applied (Section 2). Second, and this is the core. The phase of a closed path is half the solid angle enclosed──returning through three polarisers A-> B-> C-> A, the phase arg( A|C)( C|B)( B|A) agreed with -Omega/2 of the enclosed solid angle within 10^-9 for five random triangles (for instance Omega=-5.580641 gives phase +2.790320). The reverse path gives +Omega/2, which equals -(4pi-Omega)/2 modulo 2pi (Section 2). Third, enclosing a hemisphere flips the sign──turning a linear polarisation through half a turn in N steps and back (once around the equator, Omega=2pi) gives a phase of pi for every N from 3 to 384 (Section 3). Fourth, the amplitude returns but the sign does not──the surviving amplitude is cos^N(pi/N), rising from 0.125000 at N=3 to 0.987231 at N=384, yet the intensity when overlaid in phase with a reference cancels to 0.000041 at N=384 (Section 3). Fifth, finer steps converge to the half on any path──around a circle of angular radius 10 degrees near circular polarisation (Omega=0.095456), the phase moves from -0.039621 at N=6 to -0.047728 at N=1536, within 1.30x10^-7 of -Omega/2 (Section 4). Sixth, the separator is whether one measures intensity alone or the phase by interference with a reference──by intensity alone, the finer the steps the more fully the light returns; overlaid with a reference, the hemisphere’s phase pi remains as darkness. The formula is that of the same two-component vector as a spin-1/2 qubit, and shares its root with the -Omega/2 of Paper 68 (Section 5). Carrying a polarisation around a closed path shifts its phase by half the solid angle enclosed, and enclosing a hemisphere flips the sign: the same half as the Berry phase of a qubit. The phases of triangular paths made of polariser projections agreed with -Omega/2 in all five cases, and on the path that turns a linear polarisation through half a turn and back the phase was pi independently of N──the surviving amplitude returned to 0.987231 at N=384, yet overlaid with a reference the intensity cancelled to 0.000041. On a small circle too the phase converged to -Omega/2 as the steps were refined, the reverse path gave +Omega/2, and the full sphere 4pi corresponded to a phase of 2pi──what returned was the state; the phase remembers half the sky enclosed. Placed among the earlier papers──the same formula as the qubit of Paper 68 (the same root) was counted with light, on the Hopf fibration of Paper 08 and the Poincaré sphere of Paper 207. To be honest──the apparatus values are a model, neither partial polarisation nor loss was included, and the origin of the phase was left to Paper 68. On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated. Keywords: polarisation, Poincaré sphere, geometric phase, solid angle, Hopf map, interference. ----- 偏光子を何枚か通して元の偏光に戻せば、光は元に戻った、と思いやすい。本稿は、偏光の状態をポアンカレ球の点として追い、閉じた道を一周したあとに残る位相を、囲んだ立体角と比べる。新しい定理も法則も主張しない。 本稿の射程(射程注記):新しい定理も法則も主張しない──偏光の閉じた道に立体角の半分の位相が付くこと(パンチャラトナム 1956)、それがベリー位相(1984)と同じであること(ベリー 1987)、偏光の状態空間がホップ束であることは、いずれも既知である。実在の光学系を主張しない──偏光子の枚数・角度・振幅は模型の値で、どの装置の値でもない。部分偏光と損失を入れない──完全偏光の射影だけを扱い、部分偏光、吸収の違い、非断熱の効果は入れない。位相の由来に入らない──位相がホップ束のホロノミーであることの説明は論文68 の側で、ここでは光での数だけを出す。既刊との関係:論文68 は、量子ビットがホップ束であり、ベリー位相がその繊維のホロノミー -Omega/2 として測れることを示した──ここで使う二成分のジョーンズ・ベクトルはスピン 1/2 のスピノルと同じ形で、同じ関数から同じ位相が出る。数の一致ではなく式が同じなので、論文300 の意味で同根と書く。論文207 は、偏光を書く道具が三つあり、ジョーンズ・ベクトルは偏光していない光を書けないことを示した──ここではその三つ目、ポアンカレ球の上で閉じた道を追う。論文08 は、ホップ束(S^3 から S^2 への写像で、繊維は S^1)で底空間の面積と繊維の周長の積が全空間の体積になることを示した──ここでは、その底空間がポアンカレ球、繊維が偏光の位相である。論文18 は、2pi の三つの別根を分け、その一つを半球の立体角とした──ここでは半球の立体角 2pi が、位相の pi になる。加えたのは、偏光子の射影で作った閉じた道の位相を無作為の三角形で -Omega/2 と突き合わせたこと、半球の道で振幅が戻っても符号が戻らないことを数で示したこと、段を細かくしたときの収束を測ったこと、分離子を「強度か、干渉か」に置いたことである。 第一に、偏光の状態への写像はホップ写像である──二成分の複素ベクトルから作ったストークス・ベクトルは 2000 個すべてで単位球に乗り、全体の位相を掛けても 10^-15 の桁で動かない(第2節)。 第二に、これが本稿の芯である。閉じた道の位相は、囲んだ立体角の半分である──偏光子三枚で A-> B-> C-> A と戻したときの位相 arg( A|C)( C|B)( B|A) は、五つの無作為の三角形で囲んだ立体角の -Omega/2 と 10^-9 以内で一致した(例:Omega=-5.580641 で位相 +2.790320)。逆回りは +Omega/2 で、-(4pi-Omega)/2 と 2pi を法として同じである(第2節)。 第三に、半球を囲めば、符号が反転する──直線偏光の向きを N 段で半回転させて戻す(赤道を一周・Omega=2pi)と、位相は N=3 から 384 のすべてで pi である(第3節)。 第四に、振幅は戻るのに、符号は戻らない──残る振幅は cos^N(pi/N) で、N=3 の 0.125000 から N=384 の 0.987231 まで 1 に近づくのに、参照光と同位相で重ねた強度は N=384 で 0.000041 に打ち消す(第3節)。 第五に、段を細かくすると、どの道でも半分に収束する──円偏光の近くの角半径 10 度の円(Omega=0.095456)を N 段で一周すると、位相は N=6 の -0.039621 から N=1536 の -0.047728 へ、-Omega/2 に 1.30x10^-7 まで近づく(第4節)。 第六に、分離子は、強度だけを測るか、参照と干渉させて位相を測るかである──強度だけなら N を増やすほど元に戻り、参照と重ねれば半球の位相 pi が暗さとして残る。式はスピン 1/2 の量子ビットと同じ二成分ベクトルのもので、論文68 の -Omega/2 と同根である(第5節)。 偏光を閉じた道で一周させると位相は囲んだ立体角の半分だけ動き、半球を囲めば符号が反転する──量子ビットのベリー位相と同じ半分である。偏光子の射影で作った三角形の道の位相は五つとも -Omega/2 と一致し、直線偏光の向きを半回転させて戻す道では位相が N に依らず pi だった──残る振幅は N=384 で 0.987231 まで戻るのに、参照光と重ねた強度は 0.000041 に打ち消す。小さい円でも段を細かくすれば -Omega/2 に収束し、逆回りは +Omega/2 で、全球 4pi が位相の 2pi に対応する──元に戻ったのは状態で、位相は囲んだ空の半分を覚えている。既刊との位置──論文68 の量子ビットと同じ式(同根)を光で数え、論文08 のホップ束と論文207 のポアンカレ球の上に置いた。正直に言えば──装置の値は模型で、部分偏光も損失も入れず、位相の由来は論文68 に任せた。 作成にあたって:本稿の着想と内容は、著者自身の考察に基づくものです。文章の構成整理や英訳、数式の確認には AI(大規模言語モデル)の助力を得ました。最終的な内容の解釈や誤りがあれば、それらはすべて著者の責に帰します。お気づきの点があれば、ご教示いただければ幸いです。 キーワード:偏光、ポアンカレ球、幾何学的位相、立体角、ホップ写像、干渉。
⟨ ⟩ The qang project describes qubits by measurement—level quantities, the qg values qgP = P [1]. This short note states two classical results in those units, implemented and checked in the qang library (pip install qang, version 0.6.0). First, a polarization state of light is a qubit: the normalized Stokes parameters are exactly the qg values, the degree of polarization P is the length of the qg vector, and a lossless Mueller matrix is the qg gate rule qg′ = qgU†PU . Malus's law becomes I = I0(1 + qgZ cos 2θ + qgX sin 2θ)/2 for any input state. Second, the — geometric phase of a closed loop on the Bloch (Poincaré) sphere is minus half the enclosed solid angle, which is Stokes' theorem for the Berry curvature. For a loop at constant qgZ the phase is (qgZ 1)/2 of a turn: it is fixed by qgZ alone. Nothing here is new physics. The contribution is the translation into qang's units and a set of 26 numerical checks, with the phase computed three independent ways.
The Qubit as a Hopf Fibration: Is Born's "2" the Same Root as the Hopf Map's "2"? — A Disciplined Census of the Twos. The state space of a single qubit is exactly the Hopf fibration: normalised state vectors form S³⊂ℂ², the global phase is the fibre U(1)=S¹, the quotient S³/U(1)=S² is the Bloch sphere, and S³→S² is the Hopf map. The Hopf (Bloch) map (α,β)↦⟨ψ|σ|ψ⟩ is quadratic in the amplitudes, and so is Born's rule p=|ψ|². Proposition: both are values of the single sesquilinear pairing ⟨ψ|A|ψ⟩ (A Hermitian) — so Born's "2" and the Hopf map's "2" are the SAME root: the amplitude enters twice, once conjugated (and p₀=(1+n_z)/2). This is the one place where two of the system's "twos" truly coincide in origin. Applying the discipline of Papers 48 and 58 to the rest, a census shows the other twos — the spatial-flux n=2 (Papers 01, 53), the Euler characteristic χ(S²)=2 (Paper 67), the trace-reversal/ladder 2 (Paper 58), and the spinor/double-cover 2 (Berry phase = ½·solid angle; the 720° periodicity) — coincide in value without sharing an origin. The Berry phase is the Hopf fibre's holonomy, tying to the solid angle of Paper 67 while remaining a distinct spinor root. Reached through the spin/phase door; no new physics, no derivation of Born, and no use of the (distinct) mass–instanton 16π² bridge. A note on the aim of this work. This is a structural interpretation applying the distinct-roots method of Paper 58, not a proposal for a new physical law and not a derivation of the Born rule. Quantum mechanics is assumed. Nothing here is intended to contradict existing results. On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated. ----- Hopf 束としての量子ビット ── ボルン則の「2」と Hopf 写像の「2」は同根か韻か ── 規律ある「2」たちの点呼。 1個の量子ビットの状態空間は、まさに Hopf 束である:状態ベクトルは S³⊂ℂ²、全体位相が繊維 U(1)=S¹、商 S³/U(1)=S² がブロッホ球、射影 S³→S² が Hopf 写像。Hopf(ブロッホ)写像 (α,β)↦⟨ψ|σ|ψ⟩ は振幅について2次で、ボルン則 p=|ψ|² もまた2次。命題:両者はいずれも単一の双一次形式 ⟨ψ|A|ψ⟩(A エルミート)の値である——ゆえにボルン則の「2」と Hopf 写像の「2」は同根:振幅が二度、一度は共役として入る(かつ p₀=(1+n_z)/2)。これは体系の「2」たちのうち出自が真に一致する唯一の場所。論文48・58の規律を残りに適用し点呼を与えると、他の「2」——空間フラックスの n=2(論文01・53)、オイラー標数 χ(S²)=2(論文67)、トレース反転/階段の 2(論文58)、スピノル/二重被覆の 2(ベリー位相 = ½·立体角;720°周期)——は値では一致するが出自を共有しない。ベリー位相は Hopf 繊維のホロノミーで、論文67の立体角に接続する一方、別(スピノル)の根に留まる。スピン/位相の扉から達し、新物理もボルン則の導出も主張せず、(別根の)質量–インスタントンの 16π² の橋も使わない。 本稿の位置づけ。本稿は論文58の別根判定法を適用する構造的解釈であり、新たな物理法則の提案でも、ボルン則の導出でもありません。量子力学は仮定します。既存の結果と矛盾する意図はありません。 作成にあたって:本稿の着想と内容は、著者自身の考察に基づくものです。文章の構成整理や英訳、数式の確認には AI(大規模言語モデル)の助力を得ました。最終的な内容の解釈や誤りがあれば、それらはすべて著者の責に帰します。お気づきの点があれば、ご教示いただければ幸いです。 ----- Der Zustandsraum eines einzelnen Qubits ist genau die Hopf-Faserung: Normierte Zustandsvektoren bilden S³ subset of ℂ², die physikalisch belanglose globale Phase ist die Faser U(1)=S¹, und der Quotient S^3/U(1)=S² ist die Bloch-Kugel; die Projektion S³→ S² ist die Hopf-Abbildung (Mosseri–Dandoloff 2001; Urbantke 2003). Wir beobachten, dass die Hopf-(Bloch-)Abbildung (α,β)↦(ψ|σ|ψ) in den Amplituden quadratisch ist, und ebenso die Bornsche Regel p=|ψ|². Proposition: Beide sind Fälle der einzigen sesquilinearen Paarung (ψ|A|ψ) (mit A hermitesch) ── also sind Borns "2" und die "2" der Hopf-Abbildung dieselbe Wurzel: Die Amplitude geht zweimal ein, einmal konjugiert. Das ist die einzige Stelle, an der zwei der "Zweien" des Systems wirklich im Ursprung zusammenfallen. Dann wenden wir die Disziplin der Aufsätze 48 und 58 auf die übrigen Zweien an und geben eine Zählung: Die räumliche Fluss-n=2 (Aufsätze 01, 53), die Euler-Charakteristik χ(S²)=2 (Aufsatz 67), die Spurumkehr-/Leiter-2 (Aufsatz 58) und die Spinor-/Doppelüberlagerungs-2 (Berry-Phase =(1)/(2)Ω; die 720^ deg-Periodizität) werden als im Wert übereinstimmend, ohne einen Ursprung zu teilen gezeigt. Die geometrische (Berry-)Phase wird als Holonomie der Hopf-Faser identifiziert und knüpft an den Raumwinkel von Aufsatz 67 an, bleibt aber eine verschiedene (Spinor-)Wurzel. Der Wert der Zählung liegt gerade darin, dass sie nur vereinigt, was eine Wurzel teilt, und den Rest trennt. Einordnung dieses Aufsatzes: Dieser Aufsatz bietet eine strukturelle Deutung und wendet die Methode der "verschiedenen Wurzeln" aus Aufsatz 58 auf eine Familie von Faktoren zwei an, die quer durch das Yamagishi-System auftreten. Er beansprucht keine neue Physik und leitet die Bornsche Regel nicht her. Er erreicht Hopf durch die Spin-/Phasen-Tür (das Qubit), wie empfohlen, nachdem sich die Masse–Instanton-Brücke "16als verschieden erwiesen hatte (Anmerkung in Aufsatz 15). Die eine echte Vereinigung, über die berichtet wird ── dass Borns Exponent und die Hopf-(Bloch-)Abbildung die einzige Wurzel der quantenmechanischen Sesquilinearität teilen ──, ist eine bekannte strukturelle Tatsache, die hier ausdrücklich gemacht und in eine Zählung gestellt wird, die sich bewusst weigert, die übrigen Zweien zu verschmelzen. Es wird keine Behauptung aufgestellt, die der etablierten Theorie widerspricht. Schlüsselwörter: Hopf-Faserung, Qubit, Bloch-Kugel, Bornsche Regel, Berry-Phase, Sesquilinearform, verschiedene Wurzeln, strukturelle Deutung. ----- L'espace des états d'un seul qubit est exactement la fibration de Hopf : les vecteurs d'état normalisés forment S³ subset of ℂ², la phase globale, physiquement sans importance, est la fibre U(1)=S¹, et le quotient S^3/U(1)=S² est la sphère de Bloch ; la projection S³→ S² est l'application de Hopf (Mosseri–Dandoloff 2001 ; Urbantke 2003). Nous observons que l'application de Hopf (de Bloch) (α,β)↦(ψ|σ|ψ) est quadratique dans les amplitudes, et la règle de Born p=|ψ|² l'est aussi. Proposition : toutes deux sont des cas de l'unique appariement sesquilinéaire (ψ|A|ψ) (avec A hermitien) ── donc le « 2 » de Born et le « 2 » de l'application de Hopf sont la même racine : l'amplitude entre deux fois, une fois conjuguée. C'est le seul endroit où deux des « deux » du système coïncident véritablement par l'origine. Nous appliquons ensuite la discipline des Articles 48 et 58 aux autres deux et donnons un recensement : le n=2 du flux spatial (Articles 01, 53), la caractéristique d'Euler χ(S²)=2 (Article 67), le 2 du renversement de trace / de l'échelle (Article 58) et le 2 spinoriel / du revêtement double (phase de Berry =(1)/(2)Ω ; la périodicité de 720^ deg) sont montrés coïncider en valeur sans partager d'origine. La phase géométrique (de Berry) est identifiée comme l'holonomie de la fibre de Hopf et se relie à l'angle solide de l'Article 67, tout en restant une racine distincte (spinorielle). La valeur du recensement tient précisément à ce qu'il n'unifie que ce qui partage une racine et sépare le reste. Positionnement de cet article : cet article propose une interprétation structurelle et applique la méthode des « racines distinctes » de l'Article 58 à une famille de facteurs deux qui apparaissent à travers tout le système Yamagishi. Il ne revendique aucune physique nouvelle et ne dérive pas la règle de Born. Il atteint Hopf par la porte du spin et de la phase (le qubit), comme recommandé après que le pont masse–instanton « 16π² » s'est révélé distinct (note de l'Article 15). La seule unification véritable rapportée ── que l'exposant de Born et l'application de Hopf (de Bloch) partagent l'unique racine de la sesquilinéarité quantique ── est un fait structurel connu, rendu ici explicite et placé dans un recensement qui refuse délibérément de fusionner les autres deux. Aucune affirmation contredisant la théorie établie n'est faite. Mots-clés : fibration de Hopf, qubit, sphère de Bloch, règle de Born, phase de Berry, forme sesquilinéaire, racines distinctes, interprétation structurelle. ----- Lo spazio degli stati di un singolo qubit è esattamente la fibrazione di Hopf: i vettori di stato normalizzati formano S³ subset of ℂ², la fase globale, fisicamente irrilevante, è la fibra U(1)=S¹, e il quoziente S^3/U(1)=S² è la sfera di Bloch; la proiezione S³→ S² è la mappa di Hopf (Mosseri–Dandoloff 2001; Urbantke 2003). Osserviamo che la mappa di Hopf (di Bloch) (α,β)↦(ψ|σ|ψ) è quadratica nelle ampiezze, e così la regola di Born p=|ψ|². Proposizione: entrambe sono casi dell'unico accoppiamento sesquilineare (ψ|A|ψ) (con A hermitiano) ── dunque il «2» di Born e il «2» della mappa di Hopf sono la stessa radice: l'ampiezza entra due volte, una volta coniugata. È l'unico punto in cui due dei «due» del sistema coincidono davvero per origine. Applichiamo poi la disciplina degli Articoli 48 e 58 agli altri due e diamo un censimento: il n=2 del flusso spaziale (Articoli 01, 53), la caratteristica di Eulero χ(S²)=2 (Articolo 67), il 2 dell'inversione di traccia / della scala (Articolo 58) e il 2 spinoriale / del doppio rivestimento (fase di Berry =(1)/(2)Ω; la periodicità di 720^ deg) si mostrano coincidenti nel valore senza condividere un'origine. La fase geometrica (di Berry) è identificata come l'olonomia della fibra di Hopf e si collega all'angolo solido dell'Articolo 67, pur restando una radice distinta (spinoriale). Il valore del censimento sta proprio nell'unificare solo ciò che condivide una radice e nel separare il resto. Posizionamento di questo articolo: questo articolo propone un'interpretazione strutturale e applica il metodo delle «radici distinte» dell'Articolo 58 a una famiglia di fattori due che compaiono in tutto il sistema Yamagishi. Non rivendica alcuna fisica nuova e non deriva la regola di Born. Raggiunge Hopf attraverso la porta dello spin e della fase (il qubit), come raccomandato dopo che il ponte massa–istantone «16π²» si è rivelato distinto (nota dell'Articolo 15). L'unica unificazione autentica riportata ── che l'esponente di Born e la mappa di Hopf (di Bloch) condividano l'unica radice della sesquilinearità quantistica ── è un fatto strutturale noto, reso qui esplicito e collocato in un censimento ch
When the control field of a qubit, a polarization state, or a spin-21 system is swept through a level degeneracy, its direction traces an open curve on the Bloch sphere whose endpoints are antipodal, and the geodesic rule for the open-path geometric phase becomes ambiguous: infinitely many geodesics close the path, and different closures enclose different solid angles. We resolve this ambiguity in closed form. A coordinate-free monopole connection defines the open-path solid angle Ω[C] intrinsically, and displacing the degeneracy by $ε\uhat$ closes the path with enclosed solid angle $Ω(ε\uhat)=Ω[C]+2α+O(ε)$, where α is the azimuth of the transverse part of $\uhat$ measured from the principal normal of the control curve at the crossing. The identity between geometric phase and enclosed solid angle therefore holds for exactly one closing geodesic---the great circle in the osculating plane (α=0)---supplied by the curvature at the degeneracy. Berry's π invariant under reversal of the displacement and the values ±π/2 under a reflection symmetry follow as corollaries, and the pure-state limit of the finite-temperature Uhlmann phase selects the osculating-plane closure automatically, turning the heuristic closing rules of the open-path literature into a computable prescription.
FINDING: Berry phase is the holonomy of the Hopf fibration (S³→S²), connecting quantum geometric phases to classical rotation angles via SU(2) bundle structure. MATH: - Berry phase: γₙ = i∮⟨n(R)|∇_R n(R)⟩·dR = ∮_S A(R)·dR (A = Berry connection, 1-form on parameter space) - Berry curvature: Ω_μν = ∂_μ A_ν − ∂_ν A_μ = i(⟨∂_μ n|∂_ν n⟩ − ⟨∂_ν n|∂_μ n⟩) - Hopf fibration: S³ → S² with fiber S¹; SU(2) acts on S³, U(1) phase is the fiber. Berry phase = ∫_S² Ω = 2π × (Chern number) for closed surface. - Rotation angle decomposition (arXiv:2512.04481): θ_total = θ_dynamic + θ_geometric, where θ_geometric = holonomy of the Hopf connection on the disc's orientation space (SO(3) ≅ S³/ℤ₂). - Key constants: 2π (full Berry phase for spin-½ in magnetic field), π (for spin-1), and the monopole charge g = ½ (Dirac monopole strength in SU(2) → U(1) reduction). CONNECTION: - **Hopf fibration** directly encodes the golden-ratio-adjacent structure: the base S² has curvature 1, fiber S¹ has ra Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Berry phases for spin are defined for any closed loop made by a vector changing direction in three-dimensional space. A sequence of rotations moves the vector along the loop. Each rotation is around the axis perpendicular to the moving vector. The Berry phases come from the eigenvalues of the unitary operator for the product of these rotations. The angle of the product rotation is shown to be the solid angle enclosed by the loop. The proof uses the ordinary language of quantum mechanics. The product is calculated from the commutation relations for spin. A general framework is set up to define Berry phases for other transformations and states like those for rotations and spin. The integral formula is derived. Alternatives for dynamics are shown to provide different applications and interpretations of the same mathematics. An example is used to show how one Hamiltonian may be simpler than others. Adiabatic evolution is obtained in the limit as a coupling constant goes to zero, so the adiabatic changes are made by a weak perturbation that acts over a long time. The Berry phase is the same whether the coupling constant is large or small. A rationale for the definition of Berry phases is obtained from any of the alternatives for dynamics. It is particularly clear in both the limit of adiabatic evolution, where the Hamiltonian does not contain the operator that generates Berry phases, and the opposite extreme where the Hamiltonian is the operator that generates Berry phases. The latter is illustrated with an example for spin. The general definition of Berry phases and the method of calculating them are illustrated by obtaining Berry phases from Lorentz transformations. They are similar to those obtained from rotations. A vector traces a loop on a unit hyperboloid instead of the unit sphere. In place of the solid angle, the Berry phases contain the analogous measure of the area enclosed by the loop on the hyperboloid. The sign is opposite what it is for rotations. The general definition is shown to fit any unitary representation of a semisimple Lie group. The complete set of commuting operators is chosen to contain a basis for a Cartan subalgebra of the Lie algebra of generators. If a sequence of transformations by unitary operators in the group representation takes each operator in the Cartan subalgebra around a loop in the Lie algebra back to the same operator, it can be made by unitary operators in the group representation that have all the properties required for the definition of Berry phases. The loops are made by vectors moving in a real space, the Lie algebra. The motion of the vectors must maintain the lengths and angles defined by the Cartan metric of the Lie algebra. The Berry phases are determined by one loop made by one suitably chosen operator from the Cartan subalgebra.