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Geometric phases for mixed states and decoherence Kazuo Fujikawa Institute of Quantum Science, College of Science and Technology Nihon University, Chiyoda-ku, Tokyo 101-8308, Japan Received 25 September 2006; accepted 19 October 2006 Available online 4 December 2006

Abstract The gauge invariance of geometric phases for mixed states is analyzed by using the hidden local gauge symmetry which arises from the arbitrariness of the choice of the basis set deﬁning the coordinates in the functional space. This approach gives a reformulation of the past results of adiabatic, non-adiabatic and mixed state geometric phases. The geometric phases are identiﬁed uniquely as the holonomy associated with the hidden local gauge symmetry which is an exact symmetry of the Schro¨dinger equation. The puriﬁcation and its inverse in the description of de-coherent mixed states are consistent with the hidden local gauge symmetry. A salient feature of the present formulation is that the total phase and visibility in the mixed state, which are directly observable in the interference experiment, are manifestly gauge invariant. 2006 Elsevier Inc. All rights reserved. Keywords: Geometric phase,Mixed states,Gauge invariance

1. Introduction The phase is a fundamental notion in quantum mechanics [1,2], and the study of geometric phases is an attempt to understand quantum mechanics better. The notion of gauge symmetry or equivalence class plays a basic role in the analysis of geometric phases [3–18], but its origin in the theory without gauge ﬁelds was not clear. It has been recently shown that the gauge symmetry without gauge ﬁelds is identiﬁed with the hidden local gauge symmetry inherent in the second quantization, namely, the arbitrariness of the phase choice of

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K. Fujikawa / Annals of Physics 322 (2007) 1500–1517

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coordinates in the functional space [19,20]. To be speciﬁc, one has the expansion of the ﬁeld operator X ^ xÞ ¼ ^ xÞ ð1:1Þ bn ðtÞvn ðt;~ wðt;~ n

which is invariant under the simultaneous transformations, ^bn ðtÞ ! eian ðtÞ ^bn ðtÞ and vn ðt;~ xÞ ! eian ðtÞ vn ðt;~ xÞ. For the generic case, we thus have the local symmetry U = U(1) · U(1) · U(1), . . . , and in the presence of the degeneracy of basis vectors we have U = U(n1) · U(n2) · U(n3), . . . [19]. Any formulation should preserve this exact symmetry, and the basic observation is that the Schro¨dinger amplitude with wn ð0;~ xÞ ¼ vn ð0;~ xÞ is transformed under this symmetry as wn ðt;~ xÞ ! eian ð0Þ wn ðt;~ xÞ independently of t and thus the Schro¨dinger equation is invariant under this symmetry. It has been shown that both of the adiabatic [19] and non-adiabatic [21] phases are uniquely speciﬁed as the holonomy associated with this symmetry, and the adiabatic and non-adiabatic phases are treated on a completely equal footing. This formulation emphasizes that the geometric phases arise from the holonomy of the basis vectors rather than from the holonomy of the Schro¨dinger amplitudes. The basic aspects of the hidden local symmetry are summarized in Appendix A. The deﬁnitions of geometric phases in mixed states have been proposed by several authors [22,23], and it appears that the direct connection of the geometric phase with the geometrical picture of the motion of the polarization vector, for example, is lost. The gauge symmetries and associated holonomy in the geometric phases for mixed states have also been discussed [22–26], but the quantities invariant under the gauge symmetries discussed so far are non-linear (to be precise, not bi-linear) in the Schro¨dinger amplitudes, and the total phase and visibility which are directly observable in the interference experiment [23] are not invariant under the gauge symmetries discussed so far. In the present paper, we analyze the gauge invariance of geometric phases for mixed states from the point of view of the hidden gauge symmetry explained above for pure states. We discuss the geometric phases for mixed states proposed in [23], which are direct generalizations of geometric phases in pure states. The geometric phases thus deﬁned are known to be an intrinsic property of the mixed states without referring to the ancilla subsystem [24,25]. On the other hand, the geometric phases proposed in [22] are based on the ‘‘puriﬁcation’’ and crucially depend on the ancilla part [24,25]; these geometric phases may have some implications in quantum information, for example, but we forgo their analysis. Our basic observation is that an exact treatment of the Schro¨dinger equation automatically contains all the geometric phases which are uniquely identiﬁed as the holonomy associated with the hidden local symmetry, without going through an analysis of the projective Hilbert space. More importantly, the total phase and visibility which are directly observable in the interference experiment [23] are manifestly invariant under the hidden local gauge symmetry. 2. Geometric phases in mixed states 2.1. Conventional formulation In the conventional formulation of geometric phases it is customary to classify the geometric phases into adiabatic and non-adiabatic ones. In connection with the geometric

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K. Fujikawa / Annals of Physics 322 (2007) 1500–1517

phase for mixed states, we ﬁrst brieﬂy summarize the customary analysis of non-adiabatic phase which starts with [9] Z ^ ðtÞwðt;~ xÞ ¼ H xÞ; d3 x wy ðt;~ xÞwðt;~ xÞ ¼ 1 ð2:1Þ i hot wðt;~ and the cyclic evolution is deﬁned by xÞ wðT ;~ xÞ ¼ ei/ wð0;~

ð2:2Þ

with a constant /. The equivalence class of state vectors, i.e.,‘‘projective Hilbert space’’, is deﬁned by [9,12] feiaðtÞ wðt;~ xÞg;

ð2:3Þ

and we have an equivalence class of Hamiltonians ^ ðtÞ h oaðtÞg fH

ð2:4Þ

with an arbitrary function a(t) to maintain the Schro¨dinger equation. The total phase of the cyclic evolution is given by /T ¼ arg wy ð0;~ xÞwðT ;~ xÞ ¼ arg wy ð0;~ xÞUðtÞwð0;~ xÞ

ð2:5Þ

Z i t^ H ðtÞ dt UðtÞ ¼ T exp h 0

ð2:6Þ

with H

where Tw stands for the time ordering operation. Note that the total phase is not invariant under the equivalence class (2.3). One may then deﬁne the ‘‘dynamical phase’’ by Z Z 1 T ^ ðtÞwðt;~ dt d3 x wy ðt;~ xÞH xÞ /D ¼ h 0 ð2:7Þ Z T Z 3 y ¼ i dt d x w ðt;~ xÞ ot wðt;~ xÞ: 0

The non-adiabatic phase deﬁned by the diﬀerence of /T and /D Z Z 1 T y ^ ðtÞwðt;~ /G ¼ arg w ð0;~ xÞwðT ;~ xÞ þ dt d3 x wy ðt;~ xÞH xÞ h 0 Z T Z y _ ¼ arg wy ð0;~ xÞUðtÞwð0;~ xÞ þ i dt d3 x wy ð0;~ xÞUðtÞ UðtÞwð0;~ xÞ

ð2:8Þ

0

is invariant under the equivalence class (2.3) and (2.4). This is customarily called the gauge invariance condition of non-adiabatic phase. xÞ ¼ eiaðtÞ wðt;~ By using the equivalence class, one may choose a representative wðt;~ xÞ to satisfy the parallel transport condition Z Z xÞ ¼ i d3 x wy ðt;~ y ðt;~ xÞ ot wðt;~ xÞ ot wðt;~ xÞ ot aðtÞ ¼ 0 ð2:9Þ i d3 x w namely,

K. Fujikawa / Annals of Physics 322 (2007) 1500–1517

Z t Z 3 y wðt;~ xÞ ¼ exp i dt d x w ðt;~ xÞiot wðt;~ xÞ wðt;~ xÞ:

1503

ð2:10Þ

0

xÞ, which is non-linear in the Schro¨dinger amplitude, the non-adiabatic phase For this wðt;~ for the pure state is given by [14,15] y ð0;~ ;~ /G ¼ arg w xÞwðT xÞ Z y ¼ arg w ð0;~ xÞ exp i

Z

T

dt

d x w ðt;~ xÞiot wðt;~ xÞ wðT ;~ xÞ : 3

y

ð2:11Þ

0

This quantity is conﬁrmed to be invariant under the gauge transformation, and /G stands xÞ for cyclic evolution. for the holonomy of the vector wðt;~ y In the case of the mixed state described by the density matrix qðtÞ ¼ UðtÞqð0ÞUðtÞ , the interference pattern is given by [23] I / 1 þ jTr UðT Þqð0Þj cos½v arg Tr UðT Þqð0Þ

ð2:12Þ

where v stands for the variable U(1) phase in one of the interference beams. The total phase of the mixed state is thus given by cT ¼ arg Tr UðT Þqð0Þ:

ð2:13Þ

In a basis where the density matrix is written as qð0Þ ¼

N X

xk jkihkj

ð2:14Þ

k¼1

the total phase is written as cT ¼

N X

xk arg ½hkjUðT Þjki:

ð2:15Þ

k¼1

The parallel transport condition for a mixed state is imposed by requiring that Tr qðtÞUðt þ dtÞUðtÞy be real and positive which in turn leads to y y _ _ Tr qðtÞUðtÞUðtÞ ¼ Tr qð0ÞUðtÞ UðtÞ ¼0

or equivalently N h i X y _ xk hkjUðtÞ UðtÞjki ¼ 0:

ð2:16Þ

ð2:17Þ

k¼1

Under this condition, the ‘‘dynamical phase’’ deﬁned by Z Z T h i 1 T y _ ^ ðtÞ ¼ i cD ¼ dt Tr qðtÞH dt Tr qð0ÞUðtÞ UðtÞ h 0 0

ð2:18Þ

vanishes identically. It has also been asserted [23] that the condition (2.16) or (2.17) while necessary, is not suﬃcient. Instead, the stronger conditions [27] y _ hkjUðtÞ UðtÞjki ¼ 0;

k ¼ 1; 2; . . . ; N

ð2:19Þ

have been proposed; all the constituent pure states in the mixed state are required to be parallel transported independently.

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K. Fujikawa / Annals of Physics 322 (2007) 1500–1517

To understand the prescription (2.19), it is convenient to analyze the N-state density matrix in the diagonal basis. On may then consider the transformation [26] UðtÞ ! U 0 ðtÞ ¼ UðtÞ

N X

eihk ðtÞ jkihkj

ð2:20Þ

k

then the orbit of the density matrix remains unchanged qð0Þ ! q0 ðtÞ ¼ U 0 ðtÞqð0ÞU 0 ðtÞy ¼ UðtÞqð0ÞUðtÞy ¼ qðtÞ: The total phase is then transformed as cT !

c0T

0

¼ argfTr ½qð0ÞU ðT Þg ¼ arg

(

X

ð2:21Þ )

xk hkjUðT Þjkie

ihk ðT Þ

ð2:22Þ

k

while the ‘‘dynamical phase’’ is transformed as Z T 0 dt Tr ½qð0ÞU 0 ðtÞy U_ 0 ðtÞ cD ! cD ¼ i 0 Z T X _ ¼ i dt Tr ½qð0ÞUðtÞy UðtÞ xk ðhk ðT Þ hk ð0ÞÞ: þ 0

ð2:23Þ

k

The simple subtraction of the ‘‘dynamical phase’’ from the total phase does not give a result invariant under the above transformation for the general non-degenerate mixed state. To alleviate this gauge non-invariance, a manifestly gauge invariant expression ( Z T ) X y _ cG ½U ¼ arg xk hkjUðT Þjki exp i dt hkjUðtÞ iUðtÞjki ð2:24Þ 0

k

has been proposed [26]. When the stronger condition (2.19) is imposed, this expression of cG ½U agrees with the total phase. We note that the expression for cG ½U (2.24) is written as ( Z T ) X y y cG ½U ¼ arg xk wk ð0Þwk ðT Þ exp i dt wk ðtÞiot wk ðtÞ ð2:25Þ 0

k

in terms of the Schro¨dinger amplitudes deﬁned by wk ðtÞ ¼ UðtÞjki which satisfy ^ ðtÞwk ðtÞ; i h ot wk ðtÞ ¼ H

k ¼ 1 N:

ð2:26Þ

The formula for cG ½U, which is non-linear (to be precise, not bi-linear) in the Schro¨dinger amplitudes, is thus a direct generalization of (2.11) for a pure state, and it is invariant under the transformation (2.20), namely, the equivalence class feihk ðtÞ wk ðtÞg;

k ¼ 1 N:

Note that cG ½U is also written as ( Z T ) X y i y ^ ðtÞwk ðtÞ cG ½U ¼ arg xk wk ð0Þwk ðT Þ exp dt wk ðtÞH h 0 k and thus one implicitly assumes the equivalence class of Hamiltonians

ð2:27Þ

ð2:28Þ

K. Fujikawa / Annals of Physics 322 (2007) 1500–1517

^ ðtÞ fH hot hk ðtÞg;

k¼1N

1505

ð2:29Þ

for each state wk(t) separately corresponding to (2.27). The stronger condition (2.19) is written as ^ ðtÞwk ðtÞ ¼ 0; wyk ðtÞi hot wk ðtÞ ¼ wyk ðtÞH

k ¼ 1 N:

ð2:30Þ

The Schro¨dinger equation (2.26) is, however, not maintained under the gauge transformation (2.27) except for the case ot h1 ðtÞ ¼ ¼ ot hN ðtÞ:

ð2:31Þ

For this special case, the modiﬁed Schro¨dinger equation ^ 0 ðtÞw0 ðtÞ; ih ot w0k ðtÞ ¼ H k

k¼1N

ð2:32Þ

ˆ (t) = H ˆ (t) ⁄othk(t) is satisﬁed. All the with ¼e wk ðtÞ and the k-independent H pure states in the density matrix in quantum statistical mechanics, for example, are spec^ iﬁed by a single Hamiltonian, and thus the expression of the partition function Tr ebH is valid. One may impose the same constraint on the density matrix in the present case also, and thus the condition (2.31); the universal Hamiltonian for all the pure states and the gauge invariance requirement (2.27) are not compatible. The interpretation of the geometric phase (2.25) as the holonomy associated with the gauge transformation (2.27), which is a direct generalization of (2.11), is not compatible with the universal Hamiltonian. It is also signiﬁcant that the physical observable (2.12) which is expressed in terms of the total phase and visibility jTr ½qð0ÞUðT Þj is not gauge invariant as is shown in (2.22). w0k ðtÞ

ihk ðtÞ

0

2.2. A new formulation ˆ (t) and thus given We start with hermitian Hamiltonian H R t a given ^ ðtÞdt in (2.6). We employ a diagonal form of the density matrix UðtÞ ¼ T H exp hi 0 H X qð0Þ ¼ xk jkihkj; hkjli ¼ dk;l ð2:33Þ k

and observe that wyk ðtÞwl ðtÞ ¼ dk;l

ð2:34Þ

for wk ðtÞ ¼ UðtÞjki in (2.26). We then have the total phases for pure states wk(t) as /k ðtÞ ¼ arg wyk ð0Þwk ðtÞ:

ð2:35Þ

The complete set of basis vectors {vk(t)} in Appendix A may then be chosen as vk ðtÞ ¼ ei/k ðtÞ wk ðtÞ;

vyk ðtÞvl ðtÞ ¼ dk;l :

ð2:36Þ

Note that vyk ð0Þvk ðtÞ ¼ real and positive in the present deﬁnition. In the general ~ x-dependent case, we deﬁne the Schro¨dinger amplitudes by Z wk ðt;~ xÞ ¼ h~ xjUðtÞjki ¼ d3 y h~ xjUðtÞj~ yivk ð0;~ yÞ

ð2:37Þ

ð2:38Þ

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K. Fujikawa / Annals of Physics 322 (2007) 1500–1517

and vk ðt;~ xÞ ¼ ei/k ðtÞ wk ðt;~ xÞ. We can also write the Schro¨dinger amplitudes as [21] Z Z i t o ^ ðtÞvk ð~ ð2:39Þ d3 x vyk ð~ x; tÞ ¼ vk ð~ x; tÞ exp x; tÞH x; tÞ hkjih jki wk ð~ h 0 ot with hkjih

o jki ot

Z

d3 x vyk ð~ x; tÞi h

o vk ð~ x; tÞ: ot

ð2:40Þ

The Schro¨dinger amplitudes in (2.39) are non-linear in the basis vectors, but the non-linearity of the probability amplitudes in the basis vectors is common in ﬁeld theory. The existence of a hidden local symmetry in wk ðt;~ xÞ becomes clear in the second quantized formulation, where the ﬁeld operator is expanded by using the basis vectors in (2.36) X ^ xÞ ¼ ^ wðt;~ xÞ ð2:41Þ bk ðtÞvk ðt;~ k

and the eﬀective Hamiltonian in (A.4) is given by Z X o 3 y y ^ ^ ^ H eff ðtÞ ¼ d x vn ð~ x; tÞH ðtÞvn ð~ x; tÞ hnjih jni ^bn ðtÞ bn ðtÞ ot n

ð2:42Þ

^ xÞ is invariant under the hidden local gauge symmetry which is diagonal. The operator wðt;~ vk ð~ x; tÞ ! eiak ðtÞ vk ð~ x; tÞ;

^ bk ðtÞ ! eiak ðtÞ ^ bk ðtÞ

ð2:43Þ

with a general function ak(t). Any formulation should preserve this exact symmetry. The Schro¨dinger amplitude wk ðt;~ xÞ in (2.39) is transformed under the hidden local symmetry as wk ðt;~ xÞ ! eiak ð0Þ wk ðt;~ xÞ

ð2:44Þ

independently of t and thus the Schro¨dinger equation with a ﬁxed Hamiltonian is invariant under the hidden local symmetry. This property is clearly seen in the general expression (A.6) in Appendix A The diﬀerence between the equivalence class (2.27) and the present hidden local gauge symmetry is obvious. From the point of view of the hidden local symmetry, the condition (2.37) is realized by a speciﬁc choice of the hidden local gauge. The quantity Tr UðT Þqð0Þ appearing in (2.12) is then written as X Tr UðT Þqð0Þ ¼ xk wyk ð0;~ xÞwk ðT ;~ xÞ k

¼

X k

xk vyk ð0;~ xÞvk ðT ;~ xÞ exp

^ ðtÞvk ðt;~ xÞH xÞ vyk ðt;~

Z T i dt d3 x½vyk ðt;~ xÞihot vk ðt;~ xÞ: h 0 ð2:45Þ

P by using the density matrix qð0Þ ¼ k xk wk ð0;~ xÞwyk ð0;~ xÞ. This quantity is manifestly invariant under the hidden local symmetry with a ﬁxed Hamiltonian, and thus not only the total phase (2.15) but also the visibility jTr UðT Þqð0Þj in (2.12) are invariant. This expression of Tr UðT Þqð0Þ contains the holonomy vyk ð0;~ xÞvk ðT ;~ xÞ

ð2:46Þ

K. Fujikawa / Annals of Physics 322 (2007) 1500–1517

for

Z t i vk ðt;~ xÞ ¼ vk ðt;~ xÞ exp dt hkji hot jki : h 0

1507

ð2:47Þ

This holonomy is ﬁxed by the hidden local symmetry. If one considers vk ðt;~ xÞ ¼ eiak ðtÞ vk ðt;~ xÞ by using the hidden local symmetry, the parallel transport condition Z d3 xvyk ðt;~ xÞihotvk ðt;~ xÞ ¼ 0 ð2:48Þ ﬁxes ak(t) as in (2.47). If all the pure states perform cyclic evolution with the same period T, one can choose the hidden local gauge such that xÞvk ðT ;~ xÞ ¼ real and positive vyk ð0;~

ð2:49Þ

for all k, and the exponential factor R Tin (2.45) exhibits the entire geometrical phase together ^ ðtÞvk ðt;~ xÞH xÞ of each pure state. In the case with the ‘‘dynamical phase’’ ð1= hÞ 0 dt d3 x vyk ðt;~ of the cyclic evolution, one can use the formula (2.45) for the spatial interference pattern analogous to the Aharonov–Bohm phase. In practice, the cyclic evolution of all the pure states wk(t) with a period T may be rather exceptional. For a generic case, we need to deﬁne the total phase for non-cyclic evolution [12] as the phase of X Z 3 y xk d x wk ð0;~ xÞwk ðT ;~ xÞ Tr UðT Þqð0Þ ¼ k

¼

X

Z xk

3

d

x vyk ð0;~ xÞvk ðT ;~ xÞ exp

k

Z T i dt d3 x ½vyk ðt;~ xÞihot vk ðt;~ xÞ: h 0

^ ðtÞvk ðt;~ xÞH xÞ vyk ðt;~

ð2:50Þ

where we included the integration over the spatial coordinates ~ x in the trace by following the analysis in [19]. Note that (2.37), which is realized by a suitable choice of the hidden R xÞvk ðt;~ xÞ ¼ real and positive. local gauge, is written in the present notation as d3 x vyk ð0;~ A salient feature of the present formulation is that the total phase and visibility in (2.45) or (2.50), which are the direct observables in the interference experiment [23], are gauge invariant, and the geometric phase arises from the holonomy of the basis vectors rather than from the Schro¨dinger amplitudes. The Schro¨dinger equation is invariant under the hidden local symmetry with a ﬁxed Hamiltonian. The parallel transport condition on the Schro¨dinger amplitudes constrains the Hamiltonian as in (2.30), whereas the parallel transport of the basis vectors is realized without any constraint on the Hamiltonian as in (2.47). In the present formulation, the geometric phase is analyzed on the basis of a given general Hamiltonian instead of imposing constraints on the Hamiltonian, which is in accord with the original idea of the non-adiabatic phase [9,12]. For some applications in quantum information [28–32], for example, one may be interested in the inverse, namely, in ﬁnding a Hamiltonian which reproduces a given density matrix [33]. In such a case, one may impose the condition (2.30) also. In any case, if the stronger condition (2.30) happens to be satisﬁed for a speciﬁc Hamiltonian, our formula for the total phase gives the geometric phase which is the same as in the past formulation.

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K. Fujikawa / Annals of Physics 322 (2007) 1500–1517

3. Example: Spin polarization We analyze a simple example described by a time dependent Hamiltonian ^ ¼ lh~ H BðtÞ~ r; ~ BðtÞ ¼ Bðsin h cos uðtÞ; sin h sin uðtÞ; cos hÞ

ð3:1Þ

where u(t) = xt with constant x, B and h. The eﬀective Hamiltonian in (A.4) then becomes ð1 þ cos hÞ 1 cos h y ^ ^ ^ hx bþ bþ þ l hx ^by ^b H eff ðtÞ ¼ l hB hB 2 2 i sin h h^y ^ hx bþ b þ ^ by ^ bþ ð3:2Þ 2 if one uses the basis set vþ ðtÞ ¼

cos 12 heiuðtÞ sin 12 h

! ;

v ðtÞ ¼

sin 12 heiuðtÞ

! ð3:3Þ

cos 12 h

ˆ (t)v±(t) = «l⁄Bv±(t) and the relations which satisfy H o ð1 þ cos hÞ vþ ðtÞ ¼ x; ot 2 o sin h o vyþ ðtÞi v ðtÞ ¼ x ¼ vy ðtÞi vþ ðtÞ; ot 2 ot o 1 cos h x: vy ðtÞi v ðtÞ ¼ ot 2 vyþ ðtÞi

ð3:4Þ

The above eﬀective Hamiltonian equation (3.2) is not diagonal, and thus quantum state mixing takes place if one uses the basis set v±(t). We thus perform a unitary transformation ! ! ^ cos 12 a sin 12 a ^cþ ðtÞ bþ ðtÞ ¼ ^ ðtÞ ^c ðtÞ sin 12 a cos 12 a b ^cþ ðtÞ UT ð3:5Þ ^c ðtÞ where UT stands for the transpose of U. The eigenfunctions are transformed to vþ ðtÞ wþ ðtÞ ¼U w ðtÞ v ðtÞ ! cos 12 a sin 12 a vþ ðtÞ ¼ sin 12 a cos 12 a v ðtÞ

ð3:6Þ

or explicitly wþ ðtÞ ¼

cos 12 ðh aÞeiuðtÞ sin 12 ðh aÞ

! ;

w ðtÞ ¼

sin 12 ðh aÞeiuðtÞ cos 12 ðh aÞ

! ð3:7Þ

K. Fujikawa / Annals of Physics 322 (2007) 1500–1517

1509

^ xÞ in second quantization is given by The ﬁeld variable wðt;~ X ^ xÞ ¼ ^ wðt;~ bn ðtÞvn ðtÞ n¼

¼

X

^cn ðtÞwn ðtÞ

ð3:8Þ

n¼

which is invariant under the hidden local symmetry wn ðtÞ ! eian ðtÞ wn ðtÞ;

^cn ðtÞ ! eian ðtÞ^cn ðtÞ:

ð3:9Þ

We also have ^ w ðtÞ ¼ l wy ðtÞH hH cos a h x ð1 cosðh aÞÞ: hot w ðtÞ ¼ wy ðtÞi 2

ð3:10Þ

If one chooses the parameter a in (3.5) as tan a ¼

hx sin h 2l hB þ hx cos h

ð3:11Þ

or equivalently 2l⁄Bsina = ⁄xsin(h a), one obtains a diagonal eﬀective Hamiltonian [21] X hx y ^ ^cn lhB cos a H eff ðtÞ ¼ ð1 cosðh aÞÞ ^cn 2 n¼ X ^ wn ðtÞ wy ðtÞi ^cyn wyn ðtÞH ¼ hot wn ðtÞ ^cn ð3:12Þ n n¼

and thus the quantum state mixing is avoided. The above unitary transformation is timeindependent and thus the eﬀective Hamiltonian is not changed ^ eff ðby ðtÞ; b ðtÞÞ ¼ H ^ eff ðcy ðtÞ; c ðtÞÞ. We have the Schro¨dinger amplitudes with initial H conditions w±(0) = w±(0) as (see (A.7) in Appendix A) i hx ð1 cosðh aÞÞ t w ðtÞ ¼ w ðtÞ exp lhB cos a h 2 Z t i y y ^ ¼ w ðtÞ exp dt w ðtÞH w ðtÞ w ðtÞihot w ðtÞ : ð3:13Þ h 0 up to a phase by noting These expressions are periodic with a period T ¼ 2p x w±(0) = w±(T), and they are exact. From the point of view of the diagonalization of the Hamiltonian, we have not completely diagonalized the Hamiltonian since w±(t) carry certain time-dependence. These formulas w±(t) are invariant under the hidden local symmetry (3.9) up to a constant phase factor, w ðtÞ ! eia ð0Þ w ðtÞ, and the geometric phase is identiﬁed by means of the analysis of hidden local symmetry as the phase of Z i T y dt w ðtÞi hot w ðtÞ ¼ expfipð1 cosðh aÞÞg exp ð3:14Þ h 0 up to 2np. One can measure wyþ ð0Þwþ ðT Þ, for example, which is manifestly invariant under the hidden local symmetry by the interference in

1510

K. Fujikawa / Annals of Physics 322 (2007) 1500–1517 1 jw ðT Þ 2 þ

2

2

þ wþ ð0Þj ¼ jwþ ð0Þj þ Re wyþ ð0Þwþ ðT Þ ¼ 1 þ cos½ðlB cos aÞT 12Xþ

ð3:15Þ

where Xþ ¼ 2p½1 cosðh aÞ

ð3:16Þ

rwþ ðtÞ by noting stands for the solid angle subtended by wyþ ðtÞ~ wyþ ðtÞ~ rwþ ðtÞ ¼ ðsinðh aÞ cos u; sinðh aÞ sin u; cosðh aÞÞ

ð3:17Þ

¼ wy ðtÞ~ rw ðtÞ:

The separation of the non-adiabatic phase 12Xþ and the ‘‘dynamical phase’’ (lBcosa)T in (3.15) is achieved by varying the parameters in the Hamiltonian, namely, B, x and h. In passing, we note that the non-adiabatic phase 12 Xþ in (3.16) (or non-adiabatic phase in general) is topologically trivial [21] and thus the stability argument on the basis of topology is not used. To deﬁne the basis set to represent any state jkæ of the initial density matrix P qð0Þ ¼ jkihkj in (2.33), one may deﬁne linear combinations H H wþ ðtÞ þ sin w ðtÞ 2 2 i hx ~ þ ðtÞ exp l hB cos a ð1 þ cosðh aÞÞ t ¼w h 2 Z i t y ^w ~ þ ðtÞ exp ~ þ ðtÞH ~ þ ðtÞ w ~ yþ ðtÞihot w ~ þ ðtÞ ; ¼w dt w h 0 H H W ðtÞ ¼ sin wþ ðtÞ þ cos w ðtÞ 2 2 i hx ~ ðtÞ exp l hB cos a ð1 cosðh aÞÞ t ¼w h 2 Z i t y ^w ~ ðtÞ exp ~ ðtÞH ~ ðtÞ w ~ y ðtÞihot w ~ ðtÞ ¼w dt w h 0 Wþ ðtÞ ¼ cos

ð3:18Þ

where H H i ~ þ ðtÞ ¼ cos wþ ðtÞ þ sin w ðtÞ exp ½2lhB cos a þ hxðcosðh aÞÞt ; w 2 2 h H i H ~ ðtÞ ¼ sin wþ ðtÞ exp ½2l hB cos a þ hxðcosðh aÞÞt þ cos w ðtÞ: w 2 h 2 ð3:19Þ These W±(t) satisfy the Schro¨dinger equation, and one can represent any initial state jkæ by ~ ðtÞ satisfy w ~ yn ðtÞ~ choosing H suitably. The basis vectors w wm ðtÞ ¼ dn;m and ^w ~ y ðtÞH ~ ðtÞ ¼ l w hB½cos a cos H sin a sin H cos bðtÞ; ~ y ðtÞi ~ ðtÞ ¼ l hot w hB½cos að1 cos HÞ þ sin a sin H cos bðtÞ þ w

hx ð1 cosðh aÞÞ 2 ð3:20Þ

K. Fujikawa / Annals of Physics 322 (2007) 1500–1517

1511

where b(t) = [2lBcos a + xcos(h a)]t. ~ ðtÞ are not periodic in general, and they are periodic with a period T The basis set w only when T x ¼ 2pn; T ½2lB cos a þ x cosðh aÞ ¼ 2pm

ð3:21Þ

with two integers n and m. Related to this property, the geometric phase for a pure state W+(t), for example, ~ yþ ð0Þ~ Wyþ ð0ÞWþ ðT Þ ¼ w wþ ðT Þ Z i T y ^w ~ þ ðtÞH ~ þ ðtÞ w ~ yþ ðtÞihot w ~ þ ðtÞ exp dt w h 0

ð3:22Þ

arises not only from the phase factor, i.e., the second term on the exponential but also ~ yþ ð0Þ~ from the pre-factor w wþ ðT Þ for a general T which does not satisfy (3.21). One may ~ þ ðtÞ ! ei~aþ ðtÞ w ~ þ ðtÞ, under which the quantity utilize the hidden local gauge symmetry w ~ yþ ð0Þ~ (3.22) is invariant, to make the pre-factor w wþ ðT Þ real and positive, and then only the phase factor on the exponential contributes to the geometric phase. 3.1. Mixed states We study a speciﬁc density matrix deﬁned by a pure state in (3.18) qðtÞ ¼ jWþ ðtÞihWþ ðtÞj

ð3:23Þ

which is not cyclic in general. The density matrix for a mixed state may be deﬁned by H H jwþ ðtÞihwþ ðtÞj þ sin2 jw ðtÞihw ðtÞj 2 2 by assuming the de-coherence of w±(t), and qmix(t) is cyclic with a period T. The total phase may be deﬁned as the phase of [23] qmix ðtÞ ¼ cos2

ð3:24Þ

RT H H ^ ðtÞ i dt H Tr T H e h 0 qmix ð0Þ ¼ cos2 hwþ ð0Þjwþ ðT Þi þ sin2 hw ð0Þjw ðT Þi 2 2 Z H i T y ^ wþ ðtÞ wy ðtÞi ¼ cos2 wyþ ð0Þwþ ðT Þ exp dt wþ ðtÞH h o w ðtÞ t þ þ 2 h 0 Z T H i ^ ðtÞ wy ðtÞi þ sin2 wy ð0Þw ðT Þ exp dt wy ðtÞHw h o w ðtÞ t 2 h 0

ð3:25Þ which is manifestly invariant under the hidden local symmetry w ðtÞ ! eia ðtÞ w ðtÞ and one can uniquely identify the geometric phase for each pure state w±(t). In the present case, one can in principle distinguish the geometric phase from the ‘‘dynamical phase’’ contained in the total phase by varying the parameters T = 2p/x, B, h and the angle H. See also [29–31]. Note that we can in principle separate the geometric phase and the ‘‘dynamical phase’’ by varying the parameters T = 2p/x, B and h for pure states w±(t) as in (3.15). As a special example, one may choose a = p/2 in (3.11), namely 2l hB þ hx cos h ¼ 0:

ð3:26Þ

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K. Fujikawa / Annals of Physics 322 (2007) 1500–1517

^ ðtÞw ðtÞ ¼ 0 in (3.10), and only the geometric phases remain in (3.25), Then wy ðtÞH RT i H H ^ ðtÞ H h 0 dt H Tr T e qmix ð0Þ ¼ cos2 eipð1þcosðhaÞÞ þ sin2 eipð1cosðhaÞÞ ð3:27Þ 2 2 by using (3.14) and the periodicity w±(T) = w±(0). One also has ^ ðtÞw ðtÞ ¼ 0 wy ðtÞH

ð3:28Þ

and thus the parallel transport condition (2.30) in the conventional formulation is realized by a suitable choice of the Hamiltonian instead of the gauge transformation (2.27). This choice of parameters corresponds to an explicit example in [33]. In the example discussed here, the explicit form of basis vectors w±(t) in (3.7) was deﬁned by a diagonalization of the eﬀective Hamiltonian which is a natural generalization of the analysis of adiabatic phase, instead of the construction discussed in Section 2.2. The periodicity condition P w±(T) = w±(0) with T = 2p/x severely constrains the possible initial state jkæ of qð0Þ ¼ k xk jkihkj in (2.33). For generic states jkæ one may use the basis vectors ~ ðtÞ in (3.19). The basis vectors w ~ ðtÞ generally agree with the basis vectors vk(t) in (2.36) w up to hidden local gauge symmetry. 4. Discussion The mixed state generally appears as a result of de-coherence in the actual experimental situation. A precise description of de-coherent processes in quantum mechanics is involved. Operationally, one may reduce a pure state to a mixed state by random phase approximation, for example. A more elegant procedure is the puriﬁcation and its inverse, namely, reduction. See, for example, Ref. [22]. The basic idea of the reduction starts with a normalized pure (entangled) state X wðtÞ ¼ an;m jwn ðtÞij/m ðtÞi ð4:1Þ n;m

and a density matrix for the pure state qðtÞ ¼ jwðtÞihwðtÞj:

ð4:2Þ

One then takes a partial trace over the states j/m(t)æ and obtains a density matrix for a mixed state qmix ðtÞ ¼ Tr/ fqðtÞg X xn;l jwn ðtÞihwl ðtÞj ¼

ð4:3Þ

n;l

where xn;l

X

an;m aH l:m :

ð4:4Þ

m

If the dimension of the states {jwn(t)æ} is N, one may choose the dimension of the states {j/m(t)æ} to be equal to or larger than N. One then has more than N2 free parameters an,m which are naively suﬃcient to describe N2 parameters xn,l. The puriﬁcation is the inverse of the above procedure and corresponds to a construction of a pure state starting with a mixed state, though the puriﬁcation is not unique.

K. Fujikawa / Annals of Physics 322 (2007) 1500–1517

1513

The time development of the pure state is given by ^ ^ þH ~ wðtÞ ihot wðtÞ ¼ ½H

ð4:5Þ ^ ˆ acts on the states {jwn(t)æ} and H ~ on the states {j/m(t)æ}. The pure state (4.1) is where H transformed under the equivalence class feihk ðtÞ wk ðtÞg in (2.27) as X wðtÞ ! w0 ðtÞ ¼ an;m eihn ðtÞ jwn ðtÞij/m ðtÞi ð4:6Þ n;m

and thus the Schro¨dinger equation is not maintained. To satisfy the Schro¨dinger equation one needs to consider an equivalence class of Hamiltonians ^ fH hot hk ðtÞg

ð4:7Þ

for each state jwn(t)æ separately. To maintain the universal Hamiltonian for all the states contained in (4.5) only a limited set of gauge transformations (2.31) is allowed. As a consequence, one cannot achieve the parallel transport for all the states (2.30) by means of the gauge transformation, which implies that the geometric phase (2.25) is not interpreted as the holonomy associated with (2.27) if one wants to maintain the universal Hamiltonian. The limited set of gauge transformations (2.31) can achieve only (2.17). (This property is related to the general incompatibility of the equivalence class (2.27) with the superposition principle such as (4.1), as analyzed in [21].) When one adopts the diagonal form of the density matrix (2.14) it is manifestly invariant under the equivalence class as in (2.21), but one needs to use a transformed Hamiltonian to deﬁne the physical total phase in (2.22). The invariance of the density matrix does not necessarily imply the gauge invariance of physical observables. The hidden local gauge symmetry discussed in the present paper transforms the state vectors as fjwn ðtÞig ! feian ð0Þ jwn ðtÞig and thus the pure state (4.1) X wðtÞ ! w0 ðtÞ ¼ an;m eian ð0Þ jwn ðtÞij/m ðtÞi:

ð4:8Þ

ð4:9Þ

n;m

The hidden local symmetry thus maps one allowed solution of the Schro¨dinger equation to another allowed solution with a ﬁxed Hamiltonian, although the pure state itself is changed. But the diagonal density matrix and the physically observable interference in the present application are kept invariant under this transformation as in (2.45) or (2.50). In conclusion, we have illustrated the advantage of the use of the hidden local gauge symmetry, which is clearly recognized in the second quantization, in the analyses of geometric phases both for pure and mixed states. As for other applications of the second quantized formulation, it has been shown elsewhere [34] that the geometric phase and the quantum anomaly, which have been long considered to be closely related, have in fact little to do with each other. Appendix A. Hidden local gauge symmetry In this appendix, for the sake of completeness, we recapitulate the basic idea of hidden ^~ ^ X ðtÞÞ ^ ¼H ^ ð~ local gauge symmetry. We start with the generic hermitian Hamiltonian H p; x;

1514

K. Fujikawa / Annals of Physics 322 (2007) 1500–1517

for a single particle theory in the background variable X(t) = (X1(t),X2(t), . . .). We then deﬁne a complete set of (time-dependent) eigenfunctions Z d3 x vyn ð~ x; tÞvm ð~ x; tÞ ¼ dn;m ; ðA:1Þ which are arbitrary at this moment. We take the time T as a period of the variable X(t) and xÞg in the analysis of geometric phases, unless stated otherwise. the basis set fvn ðt;~ ^ xÞ in the notation of second quantization as We then expand the ﬁeld variable wðt;~ X ^ xÞ ¼ ^ wðt;~ x; tÞ: ðA:2Þ bn ðtÞvn ð~ n

By using the above expansion in the action Z ^ w ^ yH ^ ^ y ihot w ^ wg S ¼ dt d3 xfw we obtain the eﬀective Hamiltonian (depending on Bose or Fermi statistics) Z X o 3 y y ^ ^ ^ H eff ðtÞ ¼ d x vn ð~ x; tÞH vm ð~ x; tÞ hnjih jmi ^bm ðtÞ bn ðtÞ ot n;m

ðA:3Þ

ðA:4Þ

bym ðtÞ ¼ dn;m . The second term in the eﬀective Hamiltonian is deﬁned by with ½^ bn ðtÞ; ^ Z o o ðA:5Þ d3 x vyn ð~ x; tÞi h vm ð~ x; tÞ hnji h jmi: ot ot The probability amplitude which satisﬁes the Schro¨dinger equation with wn ð~ x; 0Þ ¼ vn ð~ x; 0Þ is given by ^ xÞ^ wn ð~ x; tÞ ¼ h0jwðt;~ byn ð0Þj0i

ðA:6Þ

^¼H ^ in the present problem. When one deﬁnes the Schro¨dinger picture H ^ eff ðtÞ ^w since i h ot w ˆ eﬀ(t), one can write [19,20] by replacing all ^ bn ðtÞ by ^ bn ð0Þ in the above H Z X i T ^ wn ð~ x; tÞ ¼ vm ð~ x; tÞ hmjT H exp ðA:7Þ Heff ðtÞ dt jni h 0 m where T w stands for the time ordering operation, and the state vectors in the second quantization are deﬁned by jni ¼ ^ byn ð0Þj0i. This formula is exact. Our formulation contains an exact hidden local gauge symmetry which keeps the ﬁeld ^ xÞ invariant variable wðt;~ vn ð~ x; tÞ ! v0n ð~ x; tÞ ¼ eian ðtÞ vn ð~ x; tÞ; 0 ian ðtÞ ^ ^ ^ bn ðtÞ ! b ðtÞ ¼ e bn ðtÞ; n ¼ 1; 2; 3; . . . ;

ðA:8Þ

n

where the gauge parameter an(t) is a general function of t. This gauge symmetry (or substitution rule) states the fact that the choice of coordinates in the functional space is arbitrary and this symmetry by itself does not give any conservation law. This symmetry is exact under a rather mild condition that the basis set (A.1) is not singular, and consequently the physical observables should always respect this symmetry. Our next observation is that wn ð~ x; tÞ is transformed under the hidden local gauge symmetry (A.8) as

K. Fujikawa / Annals of Physics 322 (2007) 1500–1517

w0n ð~ x; tÞ ¼ eian ð0Þ wn ð~ x; tÞ

1515

ðA:9Þ

independently of the value of t. This transformation is derived by using the exact representation (A.6) or (A.7), and it implies that wn ð~ x; tÞ is a physical object since wn ð~ x; tÞ stays in the same ray [1,2] under an arbitrary hidden local gauge transformation. A.1. Adiabatic phase We choose the speciﬁc basis set [5] h o ^ ð H ;~ x; X ðtÞÞvn ð~ x; tÞ ¼ E n ðX ðtÞÞvn ð~ x; tÞ; i o~ x

ðA:10Þ

and by assuming the dominance of diagonal elements in (A.7) in the adiabatic approximation, we have Z i t o ðA:11Þ wn ð~ x; tÞ ’ vn ð~ x; tÞ exp E n ðX ðtÞÞ hnjih jni dt : h 0 ot which is conﬁrmed to be invariant under the hidden local symmetry up to a constant phase. The product wn ð~ x; 0Þy wn ð~ x; T Þ ¼ vn ð0;~ xÞy vn ðT ;~ xÞ Z T i o exp E n ðX ðtÞÞ hnjih jni dt : h 0 ot

ðA:12Þ

is thus manifestly invariant under the hidden local symmetry. By choosing the hidden local y gauge such that vn ðT ;~ xÞ ¼ vn ð0;~ xÞ, the pre-factor vn ð0;~ xÞ vn ðT ;~ xÞ becomes real and positive, and the phase factor in (A.12) deﬁnes a physical quantity uniquely. After this gauge ﬁxing, the phase in (A.12) is still invariant under residual gauge transformations satisfying the periodic boundary condition an(0) = an(T). A.2. Non-adiabatic phase We start with the basic assumptions [9,12] Z ^ i hot wðt;~ xÞ ¼ H ðtÞwðt;~ xÞ; d3 x wy ðt;~ xÞwðt;~ xÞ ¼ 1; ~ xÞ; wðT ~ ;~ ~ xÞ; wðt;~ xÞ ¼ ei/ðtÞ wðt;~ xÞ ¼ wð0;~ /ðT Þ ¼ /; /ð0Þ ¼ 0:

ðA:13Þ

We now choose the ﬁrst element of the complete orthonormal set fvn ðt;~ xÞg in (A.1) such that ~ xÞ; xÞ ¼ wðt;~ v1 ðt;~

ðA:14Þ

xÞg is an arbitrary complete orthonormal set. The amplitude which is possible since fvm ðt;~ w1 ðt;~ xÞ in (A.7) satisﬁes the Schro¨dinger equation and w1 ð0;~ xÞ ¼ v1 ð0;~ xÞ ¼ wð0;~ xÞ: We thus have [21]

ðA:15Þ

1516

K. Fujikawa / Annals of Physics 322 (2007) 1500–1517

wðt;~ xÞ ¼ w1 ðt;~ xÞ

Z t Z i ^ v1 ðt;~ ¼ v1 ðt;~ xÞ exp dt d3 x vy1 ðt;~ xÞH xÞ h 0 Z t Z dt d3 x vy1 ðt;~ xÞi hot v1 ðt;~ xÞ

ðA:16Þ

0

where the last structure is ﬁxed by noting wðt;~ xÞ ¼ v1 ðt;~ xÞei/ðtÞ by assumtion (A.13), namely, by the assumption that only the diagonal component survives for w1 ðt;~ xÞ in (A.7). The amplitude wðt;~ xÞ is invariant under the hidden local symmetry v1 ðt;~ xÞ ! eia1 ðtÞ v1 ðt;~ xÞ up to a constant phase, wðt;~ xÞ ! eia1 ð0Þ wðt;~ xÞ. The quantity Z T Z i ^ v1 ðt;~ xÞwðT ;~ xÞ ¼ vy1 ð0;~ xÞv1 ðT ;~ xÞ exp dt d3 x vy1 ðt;~ xÞH xÞ wy ð0;~ h 0 Z T Z 3 y dt d xv1 ðt;~ xÞi hot v1 ðt;~ xÞ ðA:17Þ 0

is thus manifestly invariant under the hidden local symmetry with a ﬁxed Hamiltonian. If one chooses the gauge such that v1 ð0;~ xÞ ¼ v1 ðT ;~ xÞ as in our starting construction (A.1), the exponential factor in (A.17) extracts the entire phase from the gauge invariant quantity and, in particular, the non-adiabatic phase [9] is given by I Z b ¼ dt d3 x vy1 ðt;~ xÞi hot v1 ðt;~ xÞ: ðA:18Þ It is clear that the geometric term in (A.17) is determined uniquely by the hidden local symmetry in (A.8) without referring to any explicit form of the Hamiltonian. The basis set fvn ðt;~ xÞg specify the coordinates in the functional space, and they do not satisfy the ˆ in general. Schro¨dinger equation nor are the eigenvectors of H References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

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