1

Quantum Political Economics

Erman ZENG1

Abstract: The mathematical characterization of “the Productive Force” of a macroeconomic system is based on Newton mechanics and quantum physics, which isexpressed as the product of the growth rate of the profit rate (p) and the surplus value (M),showing several quantum qualities similar to the photon. The one-dimensional linearharmonic oscillator model can correlate the angular frequency with the change rate of therate of profit thus with the economic growth rate, leading to the quantum-like interpretationof various economic cycles. The matrix mechanics analysis of the Leontief’s input-output

table gives the Schrodinger-like value-price transformation eigen-equation ( g ),

with the reduced organic composite of capital as the eigenvalue of the price wave function,leading to the "two Cambridge controversy" resolved.Key words: quantum mechanics, harmonic oscillator, econophysics, labor theory of value,matrix operator, eigen-vector, eigen-function, eigenvalue, general equilibrium, wavefunction, transformation problem, input-output，Two Cambridge ControversyJEL：A12 E11 E25 C67 C68 D24 D46 D57 D58 E25 O47

1、Classic mechanics modelSimilar to the fractal space-time theory2, the natural logarithm of the price of a

commodity (lnP) is defined as the "displacement" (q) of the classical Newtonianphysics, the speed (v=dq/dt) corresponding to the rate of the profit (p’):

PdtdP

dtPdp

ln'

the momentum (mv) as the surplus value:M =p’Cv，Cv=C+V, C=nK:constant capital，n:turnover rate of capital K; V=wL:variable capital，w：wage rate，L：labour; thus the“productive forces3”(Φ) can be expressed as：

KLwnn

gg

CvCnnpnKpCM

nn

KCvVKwhenKnKKVnVCCv

mYgyLYgFYMpdtpdpCvp

dtdpCv

/1,M''

;)1(pM)1(

,p'M,0nKC0n:,)1()(

*)1(*)1(''''

00

00

0

，，

1 Amoy Institute of Technovation(XiaMen Productivity Promotion Center),1300 JiMei Blvd., XiaMen, FuJian

361024, PRChina, Email: zengerman@163.com2 Nottale, L. (2011)Scale Relativity and Fractal Space-Time: a NewApproach to Unifying Relativity and QuantumMechanics, Imperial College Press.3 曾尔曼，《厦门科技》4（2016）50-56.

2

Formula Φ=Φ0(n+1) reflecting the photon-like quantum character, the productivityforces(Φ) includes all the elements described by Marx4 such as the level of divisionof labor (1-), labor productivity y, number of workers L showing quantum charactertoo, the organic composition of capital g (OCC, *: general equilibrium), theproductivity coefficient F, and the Solow residual m (technical progress rate), underthe condition of the Cobb-Douglas production function5: m=F, other analogies are asfollows (Table 1)：

From the definition of annual profit rate：

)2

1(')!2

(')1!

(')1('ln

'ln'

02

2000

0

tptptiptppt

ppt

ip

ppe

ppP

epdtPdp

ii

i

ii

pt

pt

prices tend to increase, especially when p is positive; and the quantum character isdisplayed obviously。

The least action principle can be used to determine the economic equilibrium g*：

222

2

'22

''22

)'('''

''''

)(ln

pCvMpMpCvMV

vpVTL

pCvMpdtpdpM

dtpdppCv

dtdpCv

MpP

vmmaqVF

4 Karl Marx, The German Ideology, 18455 Cobb C.W., Douglas P.H. “A Theory of Production”, Amer. Econ. Rev. 1928,8(1), Spp1.139-165.6 曾尔曼，（2016）《马克思生产力经济学导引》，厦门大学出版社.(Erman ZENG,“Introducing MarxianEconomics of Productive Forces”,Xiamen Univ. Press,2016)

Table 1.Classic

MechanicsPoliticalEconomics6

mass m Cv=C+V Capital input

displacement q lnP P:price

velocity v=dq/dt p'=dlnP/dt Profit rate

acceleration a=dv/dt dp'/dt Growing rate of p’

Momentum p=mv M=p’Cv Surplus value

potential E T=p^2/2m Mp’=Ψ potential

force F=ma Φ=Mp Productive forces

3

/*,;0

)1(0

)]'2

('

[)'

(

'ln,'''

''22ln

2

VgpMCdCv

CgCVgvCCYVgpMp

MpdtdMpCv

pdtd

pL

dtd

pPdtdP

dtPdMpMp

dtdpCv

tpppCv

PL

；

According to the Lagrange equation, the "quality"of a system thus the capital inputand output (Cv=C+V) is conserved. The least action principle reveals that thegrowth rate of constant capital C equals to the growth rate of the reduced organiccomposition of capital β, the growth rate of surplus value M is equal to the growthrate of profit rate P, thus the change rate of the capital output coefficient of the CDproduction function .

In addition, according to the Noether theorem, it is shown that under thecondition of no external force, the conservation of energy is the conservation of thevalue of the commodity as well as the profit rate, unless the division of labor ischanged:

0000

)()(0'0

2/'32/'3)2/'(' 2

dQdCvvCQdCvCvdQCvQ

dtd

CvM

dtd

dtdp

dtdH

CvpMpMpMpLqpH

；

1:,1:

,)1()1(1)]1(1)[1('

1'''1'1

LR

RL

LR

ddddddp

pPppQYC

dL：division of labor，dR：roundabout production.

2、Harmonic Oscillator ModelOne dimensional linear harmonic oscillator7 model can be used to approximate

an equilibrium state of a system, the potential energy and kinetic energy are equal：

7 曾谨严，（2016,第 VI 版）《量子力学》，科学出版社.

4

RR

RR

R

RL

RRR

t

t

dddndnCnnpM

dnMnpnKpCMMp

ggdd

mddggpmdmp

KLwnn

wLnKnK

CvC

pFm

pepggdtpdp

kCvM

kMp

kKpn

kCp

kM

kMpPeP

TN

Mkp

Ck

Vk

gVk

Cvk

CvpCvkYp

T

YYM

pPdtPdp

pP

MPkpPkMp

Ntt

tWptWPCvk

PpXPmCvPXp

pCvMpHXmkXE

Cvmkkxxmamf

11

;1

'')*)((

;',',''1

/

;

'ln'ln,)1(

**1'ln

'''lnln

;4

)41(')1(1

/

111;

;0':when1or0)ln(ln0

'lnlnln

;4

,1tg

cos',sinlnln',ln,,ln'

'2

'21,)(

21

21

,

00

0

0

22

2

0

220

2

22000

；

，

）（：，

；

N is a natural number, therefore, the productive forces , “price coefficient”lnP,therate of technological progress m, various types of capitals (C, V, M) and the organiccomposition of capital g, the growth rate of profit rate (p')p , the labor divisioncoefficient dL and the production roundaboutness dR, all the parameters of theeconomic system show the quantum characters . Similar to the quantum mechanics ofthe photon quantized energy E=h and the one-dimensional harmonic oscillatorenergy En=(n+1/2)h /2 , the economic system directly reflected its quantum-likebehavior in the analytic expression of the "productive forces" =(n+1)M0；every

5

intermediate inputs (C) and output surplus value (M) are right the contribution to theproductive forces’ growth, not only in a micro commodity, but also a meso industry,and even the whole macro economy.

The development of productive forces mainly resulted from the increased ratesof both the capital turnover (n) and the production roundaboutness dR , which canreduce the capital investment (C), avoid the falling rate of profit; in other words, theacceleration of the capital turnover can offset the insufficient capital investment andlower profit rate.

Table 2 shows the calculation of the economic business cycles based on variousgrowth rates of GDP (≈p)：

Table 2 t= /4*

p= (month) year (1+4N)

0.255 47.1 3.9 50.141 84.8 7.1 90.098 122.5 10.2 130.075 160.2 13.4 170.061 197.9 16.5 210.051 235.6 19.6 250.039 311.0 25.9 330.021 574.9 47.9 610.017 725.7 60.5 770.015 801.1 66.8 850.014 838.8 69.9 890.013 951.9 79.3 1010.011 1065.0 88.8 1130.010 1178.1 98.2 1250.009 1328.9 110.7 141

After a comprehensive analysis of different economic cycles based upon hisinnovation theory, in 1936 Schumpeter8 proposed that each long cycle(the Kondratiev wave or long technological cycle of 45 to 60 years) includes 6medium cycles (the Juglar fixed-investment cycle of 7 to 11 years and the Kuznetsinfrastructural investment cycle of 15 to 25 years), each cycle includes three shortcycle (the Kitchin inventory cycle of 3 to 5 years).

Since the mid eighteenth Century, the industrial revolution has occurred threetimes. The first created the steam machine era (1760-1840, 80years), marking thetransition from agricultural civilization to industrial civilization; the second industrialrevolution moved the human being into the electric chemical age (1840-1950,110years); after the World War II, the third industrial revolution began and created anelectronic-information age (1950-2011, 61years). Since 2011, the era of artificial

8 Schumpeter, J. A. (1954). History of Economic Analysis. London: George Allen & Unwin.

6

intelligence stared . Each economic cycle with different length of the year can beexplained properly.

3. Economic Operator--Matrices MechanicsAccording to the function system of Marx's theory of value9, all the input-output

relationships are as follows：

MY

pP

MQVCebMueFunctionSurplusVal

FppwnmTheoremOkishio

LKeAVCeaVMYnctionoductionFuMarxVC

CggreducedOCC

VCgiteanicComposCapitalOrg

VCeBCQTheoryLaborValue

pt

mtFt

ft

,'',

;)1(:""

,Pr1

,1

;YCMV

10

10

10

10

：

：

；：：

：

''

,''

,''1)'1)((

pPMQ

QC

MC

pPYM

MQ

QC

YC

PppVCC

MVCC

QC

For Marx’s two Departments of production system, the Leontief intermediateinput nXn matrix is non-negative, according to the Perron-Frobenius theorem10 thereis always a positive eigenvalue , and {Pi} standing for the Marx price eigenvector：

1

1

2

2

2

2

1

1

2

2

1

121

2

2

1

2

1

2

2

2

2

1

1

1

1

212121

2

1

2

1

2

1

22

11

222221

111211

2121222222

2121111111

1,1

1,

''

''

''

''

:

''''''

1,'00'

''''

''''''

)Re:(,

QY

QC

QY

QC

QY

QC

ppp

pp

QY

QC

QY

QC

or

CYQQYCP

ppp

QQ

pp

YCYC

QpYpCpQpYpCp

QQQYYYQYCMVCproductionSimpleCYCCCQYCMVC

ii

ii

；

；

；

9 曾尔曼，（2016）《马克思生产力经济学导引》，p122,厦门大学出版社.(Erman ZENG,“Introducing MarxianEconomics of Productive Forces”,Xiamen Univ. Press,2016)10 Seneta, E. (1973) Non-negative Matrices – An Introduction to Theory and Applications. London: George Allenand Unwin.

7

112

212

21212

2121

22222

22

2112

211

12121

11

112

121

111

222

2222

2121

2

111

1212

1111

1

1

2

2

1112

2

2

1

12

2

22121

2

1

2

1

2

1

2

2

2

2

1

1

1

1

')(

-'

)(')(''')''()''(

;-ln-)(

-'

)('')''('')''(

''''''

;''''''''

''',1:1

''

'')2

''',1:1)1

1,1

1

1001

1

1

''

''

''

''

YdtppppdC

dtdp

dtdpCpp

dtdC

dtdpCp

dtdC

dtdpYQp

dtYQd

ppdtd

dtppppdY

ppdtdY

dtdp

dtdpY

dtdpCQ

dtdpYp

dtdYp

dtQCd

dtdpQp

dtdQ

dtdpYp

dtdY

dtdpCp

dtdC

dtdpQp

dtdQ

dtdpYp

dtdY

dtdpCp

dtdC

QQ

CQCpp

QY

QC

CYQpp

Pp

pp

P

P

P

P

QY

QC

QY

QC

ii

）（

）（）（

）（

；；

Based on the conservation of the value of input-outputs, the static/dynamicequilibrium conditions of Marx's simple reproduction/complex reproduction areobtained emergently. Or in a three-sector economic system--production material,living material, capital goods-- as proposed by J. Winternitz11, we also have:

ii Pp

ppp

ppp

ppp

QM

QV

QC

QM

QV

QC

QM

QV

QC

VMCCMVYQQQMVCorPPP

QQ

Q

PPP

MVCMVCMVC

MMMPMPQPMPVPCPVVVPVPQPMPVPCPCCCPCPQPMPVPCP

1,100010001

''

''

''

''

''

''

''

''

''

),(,:

,00

0000

)(:)3()(:)2()(:)1(

3

2

1

3

2

1

3

2

1

3

3

3

3

3

3

2

2

2

2

2

2

1

1

1

1

1

1

3232111321321

3

2

1

3

2

1

3

2

1

333

222

111

3213333333231

3212222232221

3211111131211

All Marx-Leontief12 input and output functions have the same form, not matter in thevalue form or in the price form:

11 Winternitz，J. ‘Values and Prices: a solution to the so-called transformation problem’, Econ. Jour. 1948, 58,276-280.12 Leontief，W. “Input-Output Economics”. 2nd ed., New York: Oxford University Press, 1986.

8

;'

''1,)''(''

'

;''1,'

''

3

1

323

1 32

3

1

3

1

3

1

3

1

i

ii

YY

i

ii

Y

iii

Y

i i

i

QQ

i i

i

Q

ii

Q

YMpVp

Pp

PM

PV

P

MV

PY

PQQ

Pp

PQ

P

Q

PQ

IPp

CvIpM

VCMr

PPIIICvdCvVCcVCc

PPIvC

IpbbVCebM

IpaBVCeaY

IpBBVCeBVCeB

PPP

QPV

PCeBQ

PQ

M

M

VCVC

M

pt

Y

Ft

Q

ftft

VC

Q

VC

ft

Q

'''

''

)1ln(,'''''

;','''';',''''

;','''''')')'('

110

101

00

10

00

10

00

10

101

10

Since the mechanics quantities can be characterized as operators, the economicinput-output eigen-equations are:

nctioniceEigenfuCCPPC

CPPPQ

CCC

PPpPQC

PQ

PQ

PPC

PQC

jiji

ijj i

ij

ii

i

ji

i

j i

ij

iii

i

jj i

ij

i

i

i

i

i

i

j j

ij

Pr:)''(11''1

'1)

''(

''

11'1

1''''

''

'

ijj i

ij

ii

i

jj i

i

i

ij

iii

i

jj i

ij

i

i

i

ii

i

i

i

jiji

i

i

j j

ij

PPCC

PpPYC

CC

PPpPYC

PY

PMV

pP

CQ

pPC

pYC

11''1

'1

''

''11

'1

''

''''

''

''

'

C’ij/C’i is the Leontief intermediate input coefficient matrix, all the eigen equations,eigenvalues and eigenvector of the price coefficient are obtained; similarly：

iii

i

jj i

ij

i

i

i

i

j j

ij

PPpPMC

PM

pPC

pMC 11

'1

'''

''

'

ii

jj ii

ij

iii

i

jj i

ij

i

i

i

i

j j

ij

PPMQC

MQC

VCCor

PPPVC

PV

PC

gVC

11''

',:

;111

1'''

1'

1

The eigenvalue (β/P’) of the eigen-equation of the intermediate input C and the total

9

output of Q demonstrated that there existed in the economic system a ratio of thereduced organic composition of capital (ROCC) and the rate of profit of P' (P '=p' +1) ,rather than the "uniform profit rate p’"13; Marx's "uniform profit rate" in the "DasKapital" should be ROCC (β), which is the eigenvalue of the eigen-equation of theintermediate input C and the total input cost Cv=C+V。

The ratio (β/p’) is the eigenvalue of the eigen-equation of the input--constantcapital C and the output--surplus value M;the ratio ( /p’) is the eigenvalue of theeigen-quation of the intermediate input C and the end use Y.Details are as following:

jjii

ij

nnn

jjii

ij

nnn

jji

ij

nn

jji

ij

nn

jji

ij

nn

jji

ij

nnnnnnn

n

n

ppMQ

C

p

pp

MQ

MQMQ

ppVC

C

p

pp

VC

VCVC

ppVC

p

pp

V

VV

pp

pYC

p

pp

Y

YY

p

pp

pMC

p

pp

M

MM

p

pP

pQC

p

pp

Q

QQ

pp

pp

CCC

CCCCCC

'''

''

''''

;''

'

''

''''

;1'

'

'

''

1

;''

'

'

''

'

;''

'

'

''

'

;''

'

'

''

'1'''

''''''

2

1

22

11

2

1

22

11

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

21

22221

11211

Wave function is the price eigen-vector：

1*

,2)()(

)(

i

ij

ij

i

jii

ppp

p

determined by price eigen-function：

13 王璐，柳欣，（2006）《马克思经济学与古典一般均衡理论》，人民出版社.

10

kiki

nnnnnnn

n

n

CC

p

pp

C

CC

p

pp

CCC

CCCCCC

'',

'

''

'''

''''''

2

1

2

1

2

1

21

22221

11211

Similar to Schrodinger function, the price wave function can be defined as：

iiji

ii

ij

ii

ij CCMQ

CVC

CgorROCCoperat

g

'',''

'''

'ˆ:

;ˆ

Similar to the intermediate input C=nK, each output--the added value Y, the surplusvalue of M, the variable capital/human capital V--displayed the quantum nature too.According to the USA 1997-2014 input-output data14, the eigenvalue (=C’/Q’) ofintermediate input C-total output Q is about 0.479 (Table 3)，however in the year of1997-2002、2004、2007、2008, the price eigenvectors are negative：

In addition, the eigen-equations can also be expressed in Dirac ket-bra notation15：

14 http://www.bea.gov/industry/io_annual.htm15 PAM Dirac (1939). "A new notation for quantum mechanics".Mathematical Proceedings of the CambridgePhilosophical Society. 35 (3): 416–418；狄拉克，（1979）《量子力学原理》，科学出版社.

11

yp

wpY

Vp

QpCPCCQQP

MQCvVp

Mp

Yp

QCPCPPCP

i jjij

1'

1'

'1'ˆ,/'

)''('1'

''

''

'1''ˆ

1

11

iinnn

nnn

n

nnnnn

n

n

n

CCCCCPCPCP

CpC

CpCCpC

PPP

p

pp

CCC

CCCCCC

PPP

''''

'

''

,,

'''

''''''

***

212211

222

111

21

2

1

21

22221

11211

21

Because the production and sale of each commodity is affected by other commodities,it is impossible to solve the "price wave function" independently like the schrodingerequation. Because all the commodities’ price are known, using the Leontiefintermediate input n x n matrix, the price eigenstate vector as the wave function areobtained. "Two Cambridge Controversy"16 about the difficulty of the aggregation ofheterogeneous capital goods in the CD production function, the “uniform profit rate”in the general equilibrium state problem, and the wage income distribution17 issueswere resolved. The wage (w) is proportional to the labor productivity (y) andinversely proportional to the rate of profit p’.

4. DiscussionThe discreteness of physical goods in the market, and the price based currencies

have the smallest unit (￥ ; $cent, penny, etc.) and the individual nature of the laborforce, is the objective basis of the discussion of the quantum nature of macroscopiceconomic system。 It is worth mentioning that the uncertainty principle in quantummechanics discovered by Heisenberg18 90 years ago, reflects that the value and theprice cannot be determined simultaneously：

16 Joseph E. Stiglitz, "The Cambridge-Cambridge Controversy in the Theory of Capital; AView from New Haven:A Review Article," Journal of Political Economy, 82(4), Jul.-Aug. 1974: 893-903.17 Erman Zeng,"From Labor Theory of Value to Price Eigen Function--Microfoundation of General Equilibrium",http://vixra.org/abs/1702.017418 Heisenberg, W. The physical content of quantum kinematics and mechanics. In Wheeler, J. A. & Zurek, W. H.(eds.) Quantum Theory and Measurement, 62–84 (Princeton UP, Princeton, NJ, 1983). [Originally published: Z.Phys., 43, 172-98 (1927)].

12

2/)(ln)'(1'/',/

2/)(ln],[ln],[1

QPppPQMMPMPipx ，

The negative value of the price eigen-vectors, corresponding to USA’s economic /financial crisis (2002/2008), need further empirical research, which may be carriedout by the Feynman path integral19 of quantum mechanics representation, to study thetime evolution the price state vector in the macro-economic system. It can provide thegreat help for the decision-making.

19 Feynman, R. P.; Hibbs, A. R. (1965). Quantum Mechanics and Path Integrals. New York: McGraw-Hill.